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Full text of "Natural philosophy;"

NATURAL PHILOSOPHY 


BY 

WILHELM OSTWALD 



TRANSLATED 
BY 

THOMAS SELTZER 



With the author** special revision for the American edition 




NEW YORK 

HENRY HOLT AND COMPANY 

1910 



"" 4 



COPYRIGHT, igio, 

BY 
HENRY HOLT AND COMPANY 



Published November, 1910 



THE QUINN * BODEN CO. 
RAHWAY N. J. 



The original of this book was published 
as volume I in Reclam s BUCHER DER 
NATURWISSENSCH AFT . 



PREFACE 

THE beginning of the twentieth century is marked 
by a sudden rise of interest in philosophy. This is 
especially manifest in the vast growth of philosophic 
literature. The present movement, it is noteworthy, 
is by no means a revival proceeding from the aca 
demic philosophy traditionally represented at the 
universities, but has rather the original character of 
natural philosophy. It owes its origin to the fact 
that after the specialization of the last half century, 
the synthetic factors of science are again vigorously 
asserting themselves. The need finally to consider 
all the numerous separate sciences from a general 
point of view and to find the connection between 
one s own activity and the work of mankind in its 
totality, must be regarded as the most prolific source 
of the present philosophic movement, just as it was 
the source of the natural philosophic endeavors a 
hundred years ago. 

But while that old natural philosophy soon ended 
in a boundless sea of speculation, the present move 
ment gives promise of permanent results, because it 
is built upon an extremely broad basis of experi 
ence. The laws of energy in the inorganic world 
and the laws of evolution in the organic world fur- 



vi Preface 

nish mental instruments for a conceptual elabora 
tion of the material provided by science, instru 
ments capable not only of unifying present knowl 
edge, but also of evoking the knowledge of the 
future. If it is not permissible to regard this 
unification as exhaustive and sufficient for all time, 
yet there is still so much left for us to do in work 
ing over the material we have on hand from the 
general points of view just mentioned, that the 
need for systematizing must be satisfied before we 
can turn our gaze upon things more remote. 

The present work is meant to serve as the first 
aid and guide in the acquisition of these compre 
hensive notions of the external world and the inner 
life. It is not meant to develop or uphold a " sys 
tem of philosophy." Through long experience as 
a teacher the writer has learned that those are the 
best pupils who soon go their own way. However, 
it is meant to uphold a certain method, that is, the 
scientific (or, if you will, the natural scientific), 
which takes its problems, and endeavors to solve its 
problems, from experience and for experience. If, 
as a result, several points of view arise that differ 
from those of the present day, and consequently de 
mand a different attitude toward important matters 
in the immediate future, this very fact affords proof 
that our present natural philosophy does not lead 
away from life, but aims to form a part of our life, 
and has a right to. 



CONTENTS 



PAGE 

INTRODUCTION i 



PART I 

GENERAL THEORY OF KNOWLEDGE n 

1. The Formation of Concepts n 

2. Science 13 

3. The Aim of Science 13 

4. Concrete and Abstract 16 

5. The Subjective Part 17 

6. Empirical Concepts 18 

7. Simple and Complex Concepts 19 

8. The Conclusion 24 

9. The Natural Laws 28 

10. The Law of Causation 31 

11. The Purification of the Causal Relation ... 34 

12. Induction 38 

13. Deduction 40 

14. Ideal Cases 44 

15. The Determinateness of Things 47 

16. The Freedom of the Will 50 

17. The Classification of the Sciences .... 53 

18. The Applied Sciences 57 



PART II 

LOGIC, THE SCIENCE OF THE MANIFOLD, AND MATHEMATICS 61 

19. The Most General Concept 61 

20. Association 63 

21. The Group 65 

vii 



viii Contents 

PAGE 

22. Negation 68 

23. Artificial and Natural Groups 69 

24. Arrangement of the Members 75 

25. Numbers 78 

26. Arithmetic, Algebra, and the Theory of Numbers 79 

27. Co-ordination 80 

28. Comparison 82 

29. Numbers 85 

30. Signs and Names 86 

31. The Written Language 89 

32. Pasigraphy and Sound Writing 92 

33. Sound Writing .96 

34. The Science of Language 97 

35. Continuity 101 

36. Measurement 107 

37. The Function 109 

38. The Application of the Functional Relation . .112 

39. The Law of Continuity 113 

40. Time and Space 118 

41. Recapitulation 124 



PART III 

THE PHYSICAL SCIENCES 127 

42. General 127 

43. Mechanics 128 

44. Kinetic Energy 132 

45. Mass and Matter 136 

46. Energetic Mechanics 138 

47. The Mechanistic Theories 140 

48. Complementary Branches of Mechanics . . . 144 

49. The Theory of Heat 147 

50. The Second Fundamental Principle .... 150 

51. Electricity and Magnetism 154 

52. Light 156 

53. Chemical Energy 159 



Contents ix 

PART IV 

PAGE 

THE BIOLOGIC SCIENCES 163 

54- Life 163 

55. The Storehouse of Free Energy 168 

56. The Soul 171 

57. Feeling, Thinking, Acting 174 

58. Society 179 

59. Language and Intercourse 182 

60. Civilization 184 

INDEX 187 



INTRODUCTION 

NATURAL science and natural philosophy are not 
two provinces mutually exclusive of each other. 
They belong together. They are like two roads 
leading to the same goal. This goal is the domina 
tion of nature by man, which the various natural 
sciences reach by collecting all the individual actual 
relations between the natural phenomena, placing 
them in juxtaposition, and seeking to discover their 
interdependence, upon the basis of which one 
phenomenon may be foretold from another with 
more or less certainty. Natural philosophy accom 
panies these specialized labors and generalizations 
with similar labors and generalizations, only of a 
more universal nature. For instance, while the 
science of electricity, as a branch of physics, deals 
with the relation of electrical phenomena to one 
another and to phenomena in other branches of 
physics, natural philosophy is not only concerned 
with the question of the mutual connection of all 
physical relations, but also endeavors to include in 
the sphere of its study chemical, biological, astro 
nomical, in short, all the known phenomena. In 
other words, natural philosophy is the most general 
branch of natural science. 



2 Introduction 

Here two questions are usually asked. First, how 
can we define the boundary line between natural 
philosophy and the special sciences, since, obviously, 
sharp lines of demarcation are out of the question? 
Secondly, how can we investigate and teach natural 
philosophy, when it is impossible for any one per 
son to master all the sciences completely, and so 
obtain a bird s-eye view of the general relations be 
tween all the branches of knowledge? To the be 
ginner especially, who must first learn the various 
sciences, it seems quite hopeless to devote himself to 
a study that presupposes a command of them. 

Since a discussion of the two questions will af 
ford an excellent preliminary survey of the work in 
hand, it will be well to consider them in detail. In 
the first place, the lack of complete and precise 
boundary lines is a general characteristic of all 
natural things, and science is a natural thing. If, 
for instance, we try to differentiate sharply between 
physics and chemistry, we are met with the same 
difficulty. So also in biology if we try to settle be 
yond the shadow of a doubt the line of separation 
between the animal and the vegetable kingdoms. 

If, despite this well-known impossibility, we con 
sider the division of natural things into classes and 
orders as by no means useless and do not discard 
it, but regard it as an important scientific work, 
this is practical proof that such classification pre 
serves its essential usefulness, even if it does not 
attain ideal definiteness. For, this imperfection 



Introduction 3 

notwithstanding, classification reaches its end, which 
is a comprehensive view, and thus a mastery, of 
the manifoldness of phenomena. For example, 
with the overwhelming majority of organic beings 
there is no doubt whether they are animals or plants. 
Similarly, most phenomena of inorganic nature can 
readily be designated as physical or chemical. For 
all such cases, therefore, the existing classification 
is good and useful. The few cases presenting dif 
ficulty may very well be considered by themselves 
wherever they occur, and we need merely take cog 
nizance of them here. It follows from this, to be 
sure, that classification will be all the better fitted 
to its purpose the less frequently such doubtful cases 
arise, and that we have an interest in repeatedly 
testing existing classifications with a view to finding 
out if they cannot be supplanted by more suitable 
ones. 

In these matters it is much the same as when we 
look upon the waves on the surface of a large body 
of water. Our first glance tells us that a number of 
waves are rolling there; and from a point giving 
us a sufficiently wide outlook, we can count them 
and gauge their width. But where is the line of 
division between one wave and the next? We un 
doubtedly see one wave following another, yet it is 
impossible for us to indicate precisely the end of 
one and the beginning of the next. Are we then to 
deduce that it is superfluous or unfeasible to desig 
nate the waves as different? By no means. On 



4 Introduction 

the contrary, in strictly scientific work we will en 
deavor to find some suitable definition of the 
boundary line between two consecutive waves. It 
may then be called an arbitrary line, and in a de 
gree arbitrary it will certainly be. But to the in 
vestigator this does not matter. What concerns 
him is, if, with the help of this definition, wave 
lengths can be unequivocally determined, and if this 
is possible, he will use the definition as suitable to 
the purposes of science, without dismissing from his 
mind the idea that possibly some other definition may 
provide an even easier or sharper determination. 
Such an one he would instantly prefer to the old 
one. 

Thus we see that these questions of classification 
are not questions of the so-called " essence " of the 
thing, but pertain merely to purely practical arrange- 
* ments for an easier and more successful mastery of 
scientific problems. This is an extremely important 
point of view, much more far-reaching than is ap 
parent here at its first application. 

As to the second objection, I will admit its valid 
ity. But here, too, we have a phenomenon appear 
ing in all branches and forms of science. Therefore 
we must familiarize ourselves with it in advance. 
Science was created by man for man s purposes, 
and, consequently, like all human achievements, 
possesses the indestructible quality of imperfection. 
But the mere fact that a successful working sci 
ence exists, with the help of which human life 



Introduction 5 

has been fundamentally modified, signifies that the 
t quality of incompleteness in human learning is no 
hindrance to its efficiency. For what science has 
once worked out always contains a portion of truth, 
hence a portion of efficiency. The old corpuscular 
theory of light, which now seems so childishly in 
complete to us, was adequate, none the less, for 
satisfactorily explaining the phenomena of reflection 
and refraction, and the finest telescopes have been 
built with its help. This is due to the true elements 
* in it, which taught us correctly to calculate the di 
rection of rays of light in reflection and refrac 
tion. The rest was merely an arbitrary acces 
sory which had to fall when new, contradictory 
facts were discovered. These facts could not have 
been taken into consideration when the theory was 
propounded, because they were not yet known. But 
when the corpuscular theory of light was replaced by 
the theory of waves of an elastic ether, geometric 
optics at first remained quite unchanged, because the 
theory of straight lines of rays could be deduced 
from the new views also, though not so easily and 
smoothly. And geometric optics was then con 
cerned with nothing but these straight lines, in no 
wise with the question of their propagation. It did 
not become clear until recently that this conception 
of straight lines of rays is incomplete, though, it is 
true, it made a first approach toward the presentation 
of actual phenomena. It fails when it comes to 
characterize the behavior of a pencil of rays of 



6 Introduction 

large aperture. The old idea of a straight line of 
rays was to be replaced by a more complex concept 
with more varied characteristics, namely, the wave- 
surface. The greater variety of this concept ren 
ders possible the presentation of the greater variety 
of the optical phenomena just mentioned. And 
from it proceed the very considerable advances that 
have been made, since the new theory was pro 
pounded, in optical instruments, especially the micro 
scope and the photographic objective, for the pur 
poses of which pencils of rays of large aperture are 
required. The astronomic objective with its small 
angle of aperture has not undergone particularly im 
portant improvements. 

Experience in every province of science is the 
same as in this. Science is not like a chain which 
snaps when only a single link proves to be weak. 
It is like a tree, or, better still, like a forest, in 
which all sorts of changes or ravages go on without 
causing the whole to pass out of existence or cease to 
be active. The relations between the various 
phenomena, once they become known, continue to 
exist as indestructible components of all future 
science. It may .come to pass, in fact, does come to 
pass very frequently, that the form in which those 
relations were first expressed prove to be imperfect, 
and that the relations cannot be maintained quite 
generally. It turns out that they are subjected to 
other influences which change them because they had 
been unknown, and which could not have been taken 



Introduction 7 

into consideration at the discovery and first formula 
tion of these relations. But no matter what changes 
science may undergo, a certain residue of that first 
knowledge will remain and never be lost. In this 
sense, a truth that science has once gained has life 
eternal, that is, it will exist as long as human science 
exists. 

Applying this general notion to our case, we have 
the following. How far and how generally at any 
given time the relations of the various phenomena 
are summed up in fixed forms, that is, in natural 
laws, will depend upon the stage attained by each 
of the special sciences. But since science has been 
in existence it has yielded a certain number of such 
general laws, and these, though they have been filed 
down a good deal in form and expression, and have 
undergone many corrections as to the limits of their 
application, nevertheless have preserved their es- 
* sence, since they began their existence in the brains 
of human investigators. The net of the relations of 
phenomena grows ever wider and more diversified, 
but its chief features persist. 

The same is true of an individual. No matter 
how limited the circle of his knowledge, it is a part 
of the great net, and therefore possesses the quality 
by virtue of which the other parts readily join it as 
soon as they reach the consciousness and knowledge 
of the individual. The man who thus enters the 
realm of science acquires advantages which may be 
compared to those of a telephone in his residence. 



8 Introduction 

If he wishes to, he may be connected with every 
body else, though he will make extremely limited 
use of his privilege, since he will try to reach only 
those with whom he has personal relations. But 
once such relations have been established, the pos 
sibility of telephone communication is simultaneously 
and automatically established. Similarly, every bit 
of knowledge that the individual appropriates will 
prove to be a regular part of the central organization, 
the entire extent of which he can never cover, though 
each individual part has been made accessible to him, 
provided he wants to take cognizance of it. 

The mere beginner in learning, therefore, when 
receiving the most elementary instruction in school, 
or from his parents, or even from his personal ex 
periences in his surroundings, is grasping one or 
more threads of the mighty net, and can grope his 
way farther along it in order to draw an increasing 
area of it into his life and the field of his activity. 
And this net has the valuable, even precious quality 
of being the same that joins the greatest and most 
comprehensive intellects in mankind to one another. 
The truths a man has once grasped he need never 
learn afresh so far as their actual content is con 
cerned, though not infrequently especially in newer 
sciences he may have to see the form of their 
presentation and generalization change. For this 
reason it is of such especial importance for each in 
dividual from the first to perceive these unalterable 
facts and realize that they are unalterable and learn 



Introduction 9 

to distinguish them from the alterable forms of their 
presentation. It is in this very regard that the in 
completeness of human knowledge is most clearly 
revealed. Time and again in the history of science 
form has been taken for content, and necessary 
changes of form a merely practical question have 
been confused with revolutionary modifications of 
the content. 

Thus, each presentation of a science has its natural 
philosophic portion. In text-books, whether ele 
mentary or advanced, the chapter on natural philoso 
phy is found usually at the beginning of the book, 
sometimes at the end, in the form of a " general 
introduction," or " general summary." In the spe 
cial works in which the latest advances of science 
are made known by the investigators, the natural 
philosophic portions are usually to be found in the 
form of theses, of principles, which are not dis 
cussed, often not even explicitly stated, but upon the 
acceptance of which depend all the special conclu 
sions that are drawn, in the case in hand, from the 
new facts or thoughts imparted. Whether at the 
beginning or at the end of the book, these most gen 
eral principles do not quite occupy the place that be 
fits them. If at the introduction of the text-book, 
they are practically devoid of content, since the facts 
they are meant to summarize are yet to be unfolded 
in the course of the presentation. If at the end, 
they come too late, since they have already been ap 
plied in numerous instances, though without refer- 



io Introduction 

ence to their general nature. The best method is 
and a good teacher always employs this method, 
whether in the spoken or the written word to let 
, the generalizations come whenever the individual 
facts imparted require and justify them. 

Thus, all instruction in natural sciences is neces 
sarily interspersed with natural philosophy, good or 
bad, according to the clearheadedness of the teacher. 
If we wish to obtain a perfect survey of a complex 
structure, as, for instance, the confusion of streets 
in a large city, we had better not try to know each 
street, but study a general plan, from which we 
learn the comparative situation of the streets. So 
it is well for us in studying a special science to look 
at our general plan, if for no other reason than to 
keep from losing our way when it may chance to 
lead through a quarter hitherto unknown. This is 
the purpose of the present work. 



PART I 
GENERAL THEORY OF KNOWLEDGE 

i. The Formation of Concepts. To the human 
mind, as it slowly awakens in every child, the 
world at first seems a chaos consisting of mere in 
dividual experiences. The only connection between 
them is that they follow each other consecutively. 
Of these experiences, all of which at first are dif 
ferent from one another, certain parts come to be 
distinguished by the fact that they are repeated 
more frequently, and therefore receive a spe 
cial character, that of being familiar. The famil 
iarity is due to our recalling a former similar ex 
perience; in other words, to our feeling that there 
is a relation between the present experience and 
certain former experiences. The cause of this phe 
nomenon, which is at the basis of all mental life, 
is a quality common to all living things, and mani 
festing itself in all their functions, while appearing 
but rarely or accidentally in inorganic nature. It 
is the quality by virtue of which the oftener any 
process has taken place in a living organism the more 
easily it is repeated. Here is not yet the place to show 
how almost all the characteristic qualities of living 

ii 



12 General Theory of Knowledge 

beings, from the preservation of the species to the 
highest intellectual accomplishments, are conditioned 
by this special peculiarity. Suffice it to say that be 
cause of this quality all those processes which are re 
peated frequently in any given living organism, as 
sume spontaneously, that is, from physiologic rea 
sons, a character distinguishing them essentially 
from those which appear only in isolated instances, 
or sporadically. 

If a living being is equipped with consciousness and 
thought, like man, then the conscioua; recollections of 
such uniform experiences form the enduring or per- 
* manent part in the sum-total of his experiences. 
Each time a complex event, like the change of 
seasons, for example, which we know from experi 
ence repeats itself each time a part of such an 
event reaches our consciousness, we are prepared 
also for the other parts that experience teaches 
are connected with it. This makes it possible 
*for us to foresee future events. What signifi 
cance the foreseeing of future events has for the 
preservation and the development of the individual 
as well as the species can only be indicated here. To 
give one instance, it is our ability to foretell the 
coming of winter with the impossibility of obtaining 
food directly during the winter that causes us to re 
frain from at once using up all the food we have and 
to preserve it for the day of need. The ability to 
foretell, therefore, becomes the foundation of the 
whole structure of economic life. 



The Aim of Science 13 

2. Science. The prophecy of future events based 
upon the knowledge of the details of recurring 
events is called science in its most general sense. 
Here, as in most cases in which language became 
fixed long before men had a clear knowledge of the 
things designated, the name of the thing is easily as 
sociated with false ideas arising either from errors 
that had been overcome or from other, still more ac 
cidental, causes. Thus, the mere knowledge of past 
events is also called science without any thought of 
its use for prophesying future events. Yet a mo 
ment s reflection teaches that mere knowledge of the 
past which is not meant to, or cannot, serve as a 
basis for shaping the future is utterly aimless knowl 
edge, and must take its place with other aimless 
activities called play. There are all sorts of plays re 
quiring great acumen and patient application, as for 
example the game of chess; and no one has the 
right to prevent any individual from pursuing such 
games. But the player for his part must not de 
mand special regard for his activity. By using his 
energies for his personal pleasure and not for a so 
cial purpose, that is, for a general human purpose, 
he loses every claim to the social encouragement of 
his activity, and must be content if only his in 
dividual rights are respected ; and that, too, only so 
long as the social interests do not suffer by it. 

3. The Aim of Science. These views are deliber 
ately opposed to a very widespread idea that science 
should be cultivated " for its own sake," and not for 



14 General Theory of Knowledge 

the sake of the benefits it actually brings or may be 
made to bring. We reply that there is nothing at all 
which is done merely " for its own sake." Every 
thing, without exception, is done for human pur 
poses. These purposes range from momentary per 
sonal satisfaction to the most comprehensive social 
services involving disregard of one s own person. 
But in all our actions we never get beyond the sphere 
of the human. If, therefore, the phrase " for its 
own sake " means anything, it means that science 
should be followed for the sake of the immediate 
pleasure it affords, that is to say, as play (as we have 
just characterized it), and in the " for-its-own-sake " 
demand there is hidden a misunderstood idealism, 
which, on closer inspection, resolves itself into its 
very opposite, the degradation of science. 

The element of truth hidden in that misunderstood 
phrase is, that in a higher state of culture it is found 
better to disregard the immediate technical applica 
tion in the pursuit of science, and to aim only for the 
greatest possible perfection and depth in the solution 
of its individual problems. Whether this is the cor 
rect method of procedure and when it is so, is solely 
a question of the general state of culture. In the 
early stages of human civilization such a demand 
is utterly meaningless, and all science is necessarily 
and naturally confined to immediate life. But the 
wider and more complex human relations become, 
the wider and surer must the ability to predict future 
events become. Then it is the function of prophesy- 



The Science of Concepts 15 

ing science to have answers ready for questions 
, which as yet have not become pressing, but which 
with further development may sooner or later be 
come so. 

In the net-like interlacing of the sciences, that is, 
of the various fields of knowledge, described in the 
introduction, we must always reckon with the fact 
that our anticipation of what kind of knowledge we 
shall next need must always remain very incomplete. 
It is possible to foresee future needs in general out 
line with more or less certainty, but it is impossible 
to be prepared for particular individual cases which 
lie on the border line of such anticipation, and which 
may sometimes become of the utmost importance and 
urgency. Therefore it is one of the most important 
functions of science to achieve as perfect an elabora- 
^tion as possible of all the relations conceivable, and 
in this practical necessity lies the foundation of the 
general or theoretical elaboration of science. 

The Science of Concepts. Here the question im 
mediately arises : how can we secure such perfection ? 
The answer to this general preliminary question of 
all the sciences belongs to the sphere of the first or 
the most general of all the sciences, a knowledge of 
which is presupposed for the pursuit of the other 
sciences. Since its foundation by the Greek philoso 
pher Aristotle it has borne the name of logic, which 
name, etymologically speaking, hints suspiciously at 
*the word, and the word, as is known, steps in where 
ideas are wanting. Here, however, we have to deal 



1 6 General Theory of Knowledge 

with the very science of ideas, to which language 
bears the relation only of a means and often an in 
adequate means to an end. We have already seen 
how, through the physiologic fact of memory, ex 
periences are found in our consciousness which are 
similar, that is, partially coinciding with one another. 
These coinciding parts are those concerning which 
we can make predictions, for the very reason that 
* they coincide in every single instance, and they alone, 
therefore, constitute that part of our experience 
which bears results and hence has significance. 

4. Concrete and Abstract. Such coinciding or re 
peated parts of similar experiences we call, as al 
ready stated, concepts. But here, too, attention must 
immediately be drawn to a linguistic imperfection, 
which consists in the fact that in such a group of 
coinciding experiences we designate by the same 
name both the isolated experience or the object of a 
special experience and the totality of all the coincid 
ing experiences; in other words, all the similar ex 
periences. Thus, horse means, on the one hand, 
quite a definite thing which for the moment forms 
an object of our experience, and, on the other, the 
totality of all possible similar objects which have 
been present in our former experiences, and which 
we shall meet in our future experiences. It is true 
that these two sorts of contents of consciousness bear 
ing the same name are distinguished also as concrete 
and abstract, and there is an inclination to attribute 
" reality " only to the first, while the other, as " mere 



The Subjective Part 17 

entities in thought," are relegated to a lesser degree 
of reality. As a matter of fact, the difference, 
though important, is of quite another kind. It is the 
difference between the momentary experience, as op 
posed to the totality of the corresponding memories 
and expectations. Hence not so much a difference 
in reality as in presence. However, our observations 
have already made it apparent that presence alone 
never yields knowledge. A necessary part of 

\ knowledge is the memory of former similar experi 
ences. For without such memory and the corre 
sponding comparison, it is quite impossible for us to 
get at those things which agree and which, therefore, 
may be predicted; and we should stand before every 
one of our experiences with the helplessness of a 
new-born babe.* 

5. The Subjective Part. We shall therefore have 
to recognize realities in abstract ideas in so far as 
they must rest upon some experiences to be at 
all intelligible to us. Since the formation of con 
cepts depends upon memories, and these may refer, 

1 according to the individual, to very different parts of 
the same experience of different individuals, con 
cepts always possess an element dependent upon the 



* Sometimes on suddenly awaking from a profound sleep 
a person finds himself for the moment deprived of his personal 
stock of memories, unable to recall where and in what cir 
cumstances he is. No one who has experienced such a condi 
tion can ever forget the terrifying sense of helplessness it 
brings. 



1 8 General Theory of Knowledge 

% individual, or a subjective element. This, however, 
does not consist in the addition by the individual 
of new parts not found in the experience, but, on the 

. contrary, in the different choice out of what is found 
in the experience. If every individual absorbed all 
parts of the experience, the individual, or subjective, 
differences would disappear. And since scientific 
experience endeavors to make the absorption of ex 
periences as complete as possible, it aims nearer and 

t nearer to this ideal by seeking to equalize the sub 
jective deficiency of the individual memory through 
the collocation of as many and as various memories 
as possible, thus filling in the subjective gaps in ex 
perience as far as possible and rendering them 
harmless. 

6. Empirical Concepts. First and uncondition 
ally those concepts possess reality which always and 

1 without exception are based on experienced facts. 
But we can easily make manifold arbitrary combina 
tions of concepts from different experiences, since 
*our memory freely places them at our disposal, and 
from such a combination we can form a new concept. 
Of course it is not necessary that our arbitrary com 
bination should also be found in our past or future 
experiences. On the contrary, we may rather ex 
pect that there could be many more arbitrary com- 

1 binations not to be found in experience than com 
binations later " confirmed " by experience. The 
former are purposeless because unreal, the latter, on 
the contrary, are of the utmost consequence because 



Simple and Complex Concepts 19 

upon them is based the real aim of knowledge, pre 
diction. The former are those which have brought 

f the very " reality " of the concepts into ill repute, 
while the latter show that the formation and the 
mutual reaction of the concepts practically constitute 
the entire content of all science. It is of the great 
est importance, therefore, to distinguish between the 
two kinds of concept combinations, and the study of 
this differentiation forms the content of that most 
general of all the sciences which we have character 
ized as logic, or, better, the science of concepts. 

7. Simple and Complex Concepts. The forma 
tion of concepts consists, as we have seen, 
in the selection of those parts of different but 
similar experiences which coincide with one an 
other and in the elimination of those that are dif 
ferent in kind. The results of such a procedure may 
vary greatly according to the number and the dif 
ference of the experiences placed in relation with 
one another. If, for example, we compare only a 
few experiences, and if, moreover, these experiences 
are very similar to one another, then the resulting 
concepts will contain very many parts that agree. 

. But at the same time they will have the peculiarity 
applicable to other experiences, since 



these, v are without some of the coinciding parts of 
that narrower circle. Thus, for example, the con 
cept which a rustic chained to the soil all his life 
has of human work does not apply to the work of 
the city man. A concept will embrace a larger num- 



2O General Theory of Knowledge 

f her of individual cases in proportion as it contains 
fewer different parts. And by systematically fol 
lowing out this thought we arrive at the conclusion 
that the concepts that are simple and have no dif- 

ferent parts at all find the widest application or are 
the most general. 

The elimination of the non-coinciding parts from 
the concept-forming experience is called abstraction. 
Obviously abstraction must be carried the farther 
the more numerous and the more varied the experi 
ences from which the concepts are abstracted, and 
the simplest concepts are the most abstract. 

By looking back over the ground just traversed, 
the less abstract ideas may also be regarded as the 
more complex in contradistinction to the simpler ones. 
Only we must guard against the error of literal in 
terpretation and not suppose that the less simple 
concepts have really been compounded of the simpler 
ones. In point of origin they actually existed first, 
since the experience contains the ensemble of all the 
parts, those which have been retained as well as those 
which have been eliminated. It is only later, by a 
characteristic mental operation, after we have 
analyzed the more complex concept, that is, after 
we have disclosed the simpler concepts existing in 
it, that we can compound it again; in other words, 
execute its synthesis. 

These relations bear a striking resemblance to 
the relations known from chemistry to exist be 
tween substances, namely, between elements and 



Simple and Complex Concepts 21 

compounds. From the chaos of all objects of ex 
perimentation (chemistry purposely limits itself to 
ponderable bodies) the pure substances are sifted 
out an operation corresponding to the formation 
of concepts. The pure substances prove to be either 
simple or compound, and the compounds are so con 
stituted that they can each be reduced to a limited 
number of simple substances. The simple sub 
stances, or elements, retain this quality of simplicity 
only until they are recalled ; that is, until it has 
been proved that they, too, can be resolved into still 
simpler elements. The same is true of the simple 
concepts. They can claim simplicity only until their 
complex nature is demonstrated. 

With all these similarities we must be extremely 
careful never to forget the differences existing 
alongside the agreements. So hereafter we shall 
make no further use of the chemical simile. It was 
brought into requisition merely in order to acquaint 
the beginner the more readily with the entire method 
of investigation by means of a more familiar field 
of thought and study. It is quite certain, however, 
that side by side with the given similarities there 
are also radical differences. Moreover, the notion 
of simple and complex concepts or " ideas " had 
been elaborated by John Locke long before chem 
istry reached its present state of clearness concern 
ing the concept of the elements. 

Nevertheless since then the relation has been com 
pletely reversed. While the study of the chemical 



22 General Theory of Knowledge 

elements has in the meantime undergone great de 
velopment, so that not only have the elements of all 
the substances coming under the observation of the 
chemist been discovered, but, inversely, many com 
pound substances have been constructed from their 
elements, not even an approach to such a develop 
ment is apparent in the study of concepts. On the 
contrary, the whole matter has remained at about 

fthe same point as that to which John Locke had 
brought it in the second half of the seventeenth 
century. This is due above all to the opinion of 
the most influential philosophers, that Aristotle s 
logic, or science of concepts, is absolutely true as 
well as exhaustive and complete, so that, at the ut 
most, what is left for later generations to do is only 
to make a change in the form in which the matter 
is presented. It is true that in more recent times 
the grave error of this view is beginning to be recog 
nized. We realize that Aristotle s logic embraces 

but a very small part of the entire field, though in 
this part he displays the greatest genius. But be 
yond this general recognition no great step forward 
has been made. Not even a provisional table of the 
elementary concepts has been propounded and ap 
plied since Locke. 

Hence in the following investigation we shall 
have to speak of the elements or the simpler parts 
of a complex concept only in the sense that these 
concept elements are simpler as compared with the 
complex concepts, but not in the sense that the 



Simple and Complex Concepts 23 

simplest or truly elementary concepts have already 
been worked out. It must be left to later in 
vestigators to find these, and it may be expected that 
the reduction of some concepts until then considered 
elementary into still simpler ones will take place 
chiefly in times of great intellectual progress. 

Complex concepts can, in the first place, be formed 
from experience, for in an empirical concept we 
meet with several conceptual component parts which 
can be separated from one another by a process of 
. abstraction, but are always found together in the 
given experiences. For example, the concept horse 
has originated from a very frequent, similarly re 
peated experience. On analysis it is found to con 
tain a vast number of other concepts, such as 
quadruped, vertebrate animal, warm-blooded, hairi 
ness, and so on. Horse, then, is obviously a com 
plex empirical concept. 

On the other hand, we can combine as many sim 
ple concepts as we please, even if we did not find 
them combined in experience, for in reality there 
is nothing to hinder us from uniting all the concepts 
provided by memory into any combinations we 
please. In this way we obtain complex arbitrary 
" concepts. 

The task of science can now be even more 
sharply defined than before by the fact that it per 
mits the construction of arbitrary concepts which in 
* circumstances to be foreseen become empirical con 
cepts. This is another expression for prediction, 



24 General Theory of Knowledge 

which we recognized as the characteristic of science. 
It goes deeper than the previous definition, because 
here the means for its realization are given. 

8. The Conclusion. First let us consider the sci 
entific import of the complex empirical concepts. 
It consists in the fact that they accustom us to the 
coexistence of the corresponding elements of a con 
cept. So that when, in a new experience, we meet 
with some of these elements together, we immedi 
ately suppose that we shall find in the same experi 
ence the other elements also which have not yet been 
ascertained. Such a supposition is called a conclu 
sion. A conclusion always exceeds the present ex 
perience by predicting an expected experience. 
Therefore, the form of a conclusion is the universal 
form of scientific predication. 

A conclusion must contain at least two concepts, 
f the one which is experienced, and the one which, on 
the basis of this experience, is expected. Every 
complex empirical concept makes such a conclusion 
possible after it has been separated into simpler con 
cepts. And the simplest case is naturally the one in 
which there are only two parts, or in which only 
two parts are taken into consideration. 

To what extent such a conclusion is valid, that is 
to say, to what extent the experience produces the 
anticipated concept, obviously depends upon the re 
ply to a very definite fundamental question. If in 
experience the union of the two parts of the concept 
occurs invariably, so that one part of the concept is 



The Conclusion 25 

never experienced unless the other part is also ex 
perienced, then there is the greatest probabil 
ity that the expected experience will also have the 
same character, and that the conclusion will prove 
valid or true. To be sure, there is no way of mak 
ing certain that the coincident occurrence of the two 
concepts, which experience has shown to be without 
exception hitherto, will continue to be so also in the 
future. For our only means of penetrating into the 
future consists in applying that conclusion from 
previous experiences to future experiences, and it 
can therefore by no means claim absolute validity. 
There are, however, different degrees of certainty, 
or, rather, probability, attaching to such a conclu 
sion. In experiences that occur but rarely the prob 
ability is that so far we have experienced only cer 
tain combinations of simple concepts, while others, 
though occurring, have not yet entered within the 
limited circle of our experience. In such a case a 
conclusion of the kind mentioned above may be 
right, but there is also some probability of its being 
false. On Lie other hand, in experiences which 
happen extremely frequently and in the most diverse 
circumstances, and in which we always find the con 
stant and unexceptional combination, the probability 
is very strong that we shall find the combination in 
future experiences also, and the probability of the 
conclusion approaches practical certainty. Of 
course, we can never quite exclude the possibility 
that new relations never as yet experienced might 



26 General Theory of Knowledge 

enter, by which the conclusion which hitherto has 
always been true would now become false, whether 
because the expectation entertained prove invalid in 
single instances or in all cases. 

It follows from this that in general, our con 
clusions will have the greater probability the more 

generally and the oftener the corresponding experi 
ences have occurred and are occurring. Such con 
cepts as are found consistently in many experiences 
otherwise different are called general concepts, and 
therefore the probability of the conclusions de- 

scribed will be the greater the more general the con 
cepts to which they refer. This obtains to such a 
degree that we feel that certain very general con 
clusions must be true always and without exception, 
and it is " unthinkable " to us that they can ever in 
any circumstances prove not valid. Such a state 
ment, however, is never anything else than a hidden 
appeal to experience. For the mere putting of the 
question, whether the conclusion can also be false, 
demonstrates that the opposite of what has proved 
to be the experience so far can be conceived, and the 
assertion of its " unthinkability " only signifies that 

- such an experience cannot be evoked in the mind by 
the memory for the very reason that, as has been 
premised, there are no such memories because the 
experiences did not exist. But since, on the other 
hand, there is no hindrance to thinking any combina 
tions of concepts at will, we have not the least dif 
ficulty, as everybody knows, in thinking any sort 



The Conclusion 27 

of " nonsense " whatsoever. Only it is impossible 
to reproduce such combinations from memory. 

The scientific conclusion, therefore, first takes the 
form: if A is, then B is also. Here A and B rep 
resent the two simple concepts which are known 
from experience to be found together in the more 
complex concept C. The word " is " signifies here 
some empirical reality corresponding to the concepts. 
The conclusion may therefore also be expressed, 
somewhat more circumstantially and more precisely, 
in this form: if A is experienced, the experience of 
B is also expected. The evoking of this expecta 
tion, which implies its justification, is due to the 
recollection of the coincident* of the two concepts 
in former experiences, and the probability depends, 
in the manner described above, upon the number of 
valid cases. Here it must be observed that even in 
dividual cases in which our expectations have been 
deceived do not for the most part lead us to regard 
the conclusion as generally untrue, that is, to aban 
don the expectation of B from A. For we know 
that our experience is always incomplete, that in 
certain circumstances we fail to notice existing fac 
tors, and that, therefore, our failure to find that 
relation valid which, on other occasions, has been 
found to be valid, may be attributed to subjective 
causes. In case, however, of the repeated occur 
rence of such disappointments, we will look else 
where for relations between these and other ele 
ments of experience, in order that thereafter we may 



28 General Theory of Knowledge 

foresee such cases also and include them in our an 
ticipations. 

9. The Natural Laws. The facts just described 
have very frequently found expression in the 
doctrine of the laws of nature, in connection with 
which we have often, as in the man-made social or 
political laws, conceived of a lawmaker, who, for 

* some reasons, or perhaps arbitrarily, has ordained 
that things should be as they are and not otherwise. 
But the intellectual history of the origin of the laws 
of nature shows that here the process is quite a dif 
ferent one. The laws of nature do not decree what 

, shall happen, but inform us what has happened and 
what is wont to happen. The knowledge of these 
laws, therefore, makes it possible for us, as I have 
emphasized again and again, to foresee the future 
in a certain degree and, in some measure, also to 
determine it. We determine the future by con 
structing those relations in which the desired re 
sults appear. If we cannot do so either because of 
ignorance or because of inaccessibility to the re 
quired relations, then we have no prospect of fash 
ioning the future according to our desires. The 
wider our knowledge of the natural laws, that is, 
of the actual behavior of things, the more likely and 
more numerous the possibilities for fashioning the 
future according to our desires. In this way science 

i can be conceived of as the study of how to become 

happy. For he is happy whose desires are fulfilled. 

In this conception the natural laws indicate what 



The Natural Laws 29 

simpler concepts are found in complex concepts. 
The complex concept water contains the simpler 
ones liquid, a certain density, transparency, color* 
lessness* and many others. The sentences, water 
is a liquid, water has a density of one, water is 
transparent, water is colorless, or, pale blue, etc., are 
so many natural laws. 

Now what predictions do those natural laws en 
able us to make? 

They enable us to predict that when we have 
recognized a given body as water by virtue of the 
above properties, we are justified in expecting to find 
in the same body all the other known properties of 
water. And so far experience has invariably con 
firmed such expectations. 

Furthermore, we may expect that if in a given 
specimen of water we discover a relation which 
up to that time was unknown, we shall find this 
relation also in all the other specimens of water 
even though they were not tested for that par 
ticular relation. It is obvious how enormously 
this facilitates the progress of science. For it 
is only necessary to determine this new relation 
in some one case accessible to the investigator 
to enable us to predict the same relation in all the 
other cases without subjecting them to a new test. 
As a matter of fact, this is the general method that 
science pursues. It is this that makes it possible 

* More precisely, a very pale blue. 



30 General Theory of Knowledge 

for science to make regular and generally valid 
progress through the labors of the most various in 
vestigators who work independently of one another, 
and often know nothing of one another. 

Of course, it must not be forgotten that such con 
clusions are always obtained in accordance with the 
following formula: things have been so until now, 
* therefore we expect that they will be so in the future. 
In every such case, therefore, there is the possibility 
of error. Thus far, whenever an expectation was 
not realized, it was almost always possible to find an 
" explanation " for the error. Either the inclusion 
of the special case in the general concept proved to 
be inadmissible because some of its other character 
istics were absent, or the accepted characterization 
of the concept required an improvement (limitation 
or extension). In other words, one way or an 
other, there was a discrepancy between the concept 
and the experience, and, as a rule, sooner or later 
it becomes possible for us to arrive at a better adjust 
ment between them. 

This general truth has often been interpreted to 
mean that in the end such an adjustment must of 
necessity always be possible to reach, without ex 
ception; in other words, that absolutely every part 
\ of an experience can be demonstrated as condi 
tioned by natural law. Evidently such an asser 
tion far exceeds the demonstrable. And even the 
usual conclusion cannot be applied here, that be 
cause it has happened so in the past it will happen 



The Law of Causation 31 

so in the future also. For the part of our experi 
ences that we can grasp by natural laws is in- 
finitesimally small in comparison with that in which 
our knowledge still fails us entirely. I will men 
tion only the uncertainty in predicting the weather 
for only one day ahead. Moreover, when we con 
sider that until now only the easiest problems had 
been solved, and naturally so, because they were 
most accessible to the means at hand, then we can 
readily see that experience offers no basis whatever 
for such a conclusion. We must not say, therefore, 
that because we have been able so far to explain all 
experiences by natural laws it will be so in the 
future likewise. For we are far from being able to 
explain all experiences. In fact, it is only a very 
small part that we have begun to investigate. We 
are as little justified in saying that we have ex 
plained all the problems of our experience that have 
been subjected to scientific investigation. We have 
by no means explained all of them. Every science, 
even mathematics, teems with unsolved problems. 
So we must resign ourselves to the present status of 
human knowledge and ability, and may at best ex 
press the hope founded upon previous experience, 
that we shall be able to solve more and more of the 
incalculable number of problems of our experience 
without indulging in any illusions as to the perfec 
tion of this work. 

10. The Law of Causation. By reason of its fre 
quency and importance the mental process above de- 



32 General Theory of Knowledge 

scribed has been subjected to the most diverse in 
vestigations, and that most general form of the sci 
entific conclusion (which we apply in ordinary life 
even much more frequently than in science) has 
been raised, under the name of the law of causation, 
to a principle anteceding all experience and to 
the very condition making experience possible. Of 
this so much is true, that through the peculiar physi 
ological organization of man, memory in the most 
general sense the easier execution of such processes 
as have already repeatedly taken place in the organ 
ism, as against entirely new kinds of processes 
the formation of concepts (of the recurring parts in 
the constantly changing variety of processes), is 
especially stimulated and facilitated. By it the re 
curring parts of experience step into the foreground, 
and on account of their paramount practical im 
portance for the security of life, it may well be said 
in the sense of the theory of evolution and adapta 
tion, that the entire structure and mode of life of the 
organism, especially of the human organism, nay, 
perhaps life itself, is indissolubly bound up with that 
foresight and, therefore, with the law of causation 
also. Of course, there is nothing in the way of 
calling such a relation an a priori relation, if it is 
so desired. As far as the individual is concerned it 
no doubt antedates all his experience, since the en 
tire organization which he inherits from his par 
ents had already been formed under such an in 
fluence. But that there can be forms or existence 



The Law of Causation 33 

without such an attribute is shown by the whole 
world of the inorganic, in which, as far as our 
knowledge goes, there is no evidence of either 
memory or foresight, but only of an immediate 
passive participation in the processes of the world 
around them.* 

Further, the circumstance that the causal relation 
is brought about by the peculiar manner in which 
we react upon our experiences, has sometimes been 
expressed in this way the relation of cause and ef 
fect does not exist in nature at all, but has been in 
troduced by men. The element of truth in this is, 
that a quite differently organized being, it is to be 
supposed, would be able to, or would have to, ar 
range its experiences according to quite different 
mutual relations. But since we have no experience 
of such a being, we have no possibility of forming 
a valid opinion of its behavior. On the other hand, 
we must recognize that it is possible, at least for 
mally, to conceive also of kinds of experiences with 
no coinciding parts, or a world in which there are 
no experiences at all with coinciding parts. In such, 
therefore, prediction is impossible. Such a world 
will not call forth, even in a being endowed with 
memory, a conception and generalization of the 

* It cannot be objected that inorganic nature also is known to 
be subject to the law of causation. The causal mode of re 
garding inorganic phenomena is a distinctly human one, and 
nothing justifies the assertion that the same phenomena cannot 
be viewed in an entirely different manner. 



34 General Theory of Knowledge 

various experiences in the shape of natural laws. 
Consequently we must recognize that in addition to 
the subjective factor in the formation of our knowl 
edge of the world, or that factor which is dependent 
upon our physico-psychical structure, there is also 
the objective character of the world with which we 
must decidedly reckon, or that character which is in 
dependent of us; and that in so far the natural 
laws contain also objective parts. To represent the 
relation clearly to our minds by a figure, we may 
compare the world to a heap of gravel and man to a 
pair of sieves, one coarser than the other. As 
gravel passes through the double sieve pebbles of 
apparently equal size accumulate between the sieves, 
the larger ones being excluded by the first sieve and 
the smaller ones allowed to pass by the second. It 
would be an error to assert that all the gravel con 
sisted of such pebbles of equal size. But it would 
be equally false to assert that it was the sieves that 
made the pebbles equal. 

ii. The Purification of the Causal Relation. If 
by experience we have found a proposition of the 
content, If A is, then B is also, the two concepts A 
and B generally consist of several elements which 
we will designate as a, a , a", a" , etc., and as b, 
b , b", b" . Now the question arises, whether or 
not all these elements are essential for the relation 
in question. It is quite possible, in fact, even highly 
probable, that at first only a special instance of the 
existing phenomena was found, that is, that the 



Purification of the Causal Relation 35 

concept A, which has been found to be connected 
with the concept B, contains other determining parts 
which are not at all requisite to the appearance of B. 

The general method of convincing oneself of this 
is by eliminating one by one the component parts of 
the concept A, namely, a, a , a", etc., and then seeing 
whether B still appears. It is not always easy to 
carry out this process of elimination. Our greater 
or less ability to conduct such investigations depends 
upon whether we deal with things that are merely 
the objects of our observation, and which we our 
selves have not the power to change (as, for ex 
ample, astronomical phenomena), or with things 
which are the objects of our experimentation, and 
which we can influence. In the latter case one or 
another factor is usually found which can be elim 
inated without the disappearance of B, and then we 
must proceed in such a way as to form a correspond- 
ing new concept A from the factors recognized as 
necessary (which new concept will be more general 
than the former A), and to express the given 
proposition in the improved form: If A is, then B 
is also. 

Quite similar is the case with the other member of 
this relation. It often happens that when a, or a", 
a" is found, somewhat different things appear, which 
do not fit the concept as first constructed. Then 
we must multiply the experiences as much as pos 
sible in order to determine what constant elements 
are found in the concept B, and to form from these 



36 General Theory of Knowledge 

constant elements the corresponding concept B . 
The improved proposition will then read: if A is, 
then B is also. 

This entire process may be called the purification 
of the causal relation. By this term we express the 
general fact that in first forming such a regular 
connection, the proper concepts are very seldom 
brought into relation with one another at once. 
The cause of it is that at first we make use of exist- 

* ing concepts which had been formed for quite a dif 
ferent purpose. It must therefore be regarded as 
a special piece of good fortune if these old concepts 
should at once prove suited to the new purpose. 
Furthermore, the existing concepts are as a rule so 
vaguely characterized by their names, which we must 

employ to express the new relation, that for this rea 
son also it is often necessary to determine empirically 
in what way the concept is to be definitely estab 
lished. 

The various sciences are constantly occupied with 
this work of the mutual adaptation of the concepts 
that enter into a causal relation. By way of ex 
ample, we may take the " self -understood " proposi 
tion which we use when we call out to a careless 
child when it sticks its finger into the flame of a 
candle, " Fire burns ! " We discover that there are 
self-luminous bodies which produce no increase of 
temperature, and therefore no sensation of pain. 
We discover that there are processes of combustion 
that develop no light, but heat enough to burn one s 



Purification of the Causal Relation 37 

fingers. And, finally, the scientific investigation of 
this proposition arrives at the general expression 
that, as a rule, chemical processes are accompanied 
by the development of heat, but that, conversely, 
such processes may also be accompanied by the ab 
sorption of heat. In this way that casual sentence 
which we call out to the child develops into the ex 
tensive science of thermo-chemistry when it is sub 
jected to the continuous purification of the causal 
relation, which is the general task of science. 

It remains to be added that in this process of 
adapting concepts it is necessary also sometimes to 
follow the opposite course. This is the case when 
exceptions are noticed in a relation as expressed for 
the time being; when, therefore, the proposition if 
A is present, then B is present also, is in a great 
many instances valid, but occasionally fails. This 
is an indication that in the concept A an element is 
still lacking. This element, however, is present in 
the instances that tally, but absent in the negative 
cases, and its absence is not noticed because it is not 
contained in A. Then it is necessary to seek this 
part, and after it has been found, to embody it in 
the concept A, which thus passes into the new con 
cept A . 

This case is the obverse of the former one. Here 
the more suitable concept proves to be less general 
than the concept accepted temporarily, while in the 
first case the improved concept is more general. 
Hence we formulate .the rule : exceptions to the 



38 General Theory of Knowledge 

temporary rule require a limitation, while an un 
foreseen freedom requires an extension, of the ac 
cepted concept. 

12. Induction. The form of conclusion previ 
ously discussed, because it has been so, I expect it will 
continue to be so in the future, is the form through 
which each science has arisen and has won its real 
content, that is, its value for the judgment of the 
future. It is called inference by induction, and the 

1 sciences in which it is preponderatingly applied are 
called inductive sciences. They are also called ex- 

< periential or empirical sciences. At the basis of this 
nomenclature is the notion that there are other 
sciences, the deductive or rational sciences, in which 
a reverse logical procedure is applied, whereby from 
general principles admitted to be valid in advance, 
according to an absolutely sure logical process, con 
clusions of like absolute validity are drawn. At the 
present time people are beginning to recognize the 

, fact that the deductive sciences must give up these 

claims one by one, and that they already have given 
them up to a certain extent; partly because on closer 
study they prove to be inductive sciences, and partly 
because they must forego the title and rank of a 
science altogether. The latter alternative applies 
especially to those provinces of knowledge which 
have not been used in prophesying the future or can 
not be so used. 

To return to the inductive method it is to be 
noted that Aristotle, who was the first to describe it, 



Induction 39 

proposed two kinds of induction, the complete and 
the incomplete. The first has this form : since all 
things of a certain kind are so, each individual thing 
is so. While the incomplete induction merely says : 

, since many things of a certain kind are so, pre 
sumably all things of this kind are so. One in 
stantly perceives that the two conclusions are es 
sentially different. The first lays claim to afford 
an absolutely certain result. But it rests upon the 
assumption that all the things of the kind in ques 
tion are known and have been tested as to their 
behavior. This hypothesis is generally impossible 
of fulfilment, since we can never prove that there are 
not more things of the same kind other than those 
known to us or tested by us. Moreover, the con- 

elusion is superfluous, as it merely repeats knowl 
edge that we have already directly acquired, since 
we have tested all the things of the one kind, 
hence the special thing to which the predication 
refers. 

On the other hand, the incomplete induction af 
firms something that has not yet been tested, and 

therefore involves as a condition an extension of our 
knowledge, sometimes an extremely important ex 
tension. To be sure, it must give up the claim to 
unqualified or absolute validity, but, to compensate, 
it acquires the irreplaceable advantage of lending it 
self to practical application. Indeed, in accordance 
with the scientific practice justified by experience, 
described on p. 29, the scientific inductive conclusion 



40 General Theory of Knowledge 

assumes the form : because it has once been found to 
be so, it will always be so. From this appears the 
significance of this method for the enlargement of 
science, which, without it, would have had to proceed 
at an incomparably slower pace. 

13. Deduction. In addition to the inductive 
method, science has (p. 38) another method, which, 
in a sense, should be the reverse of the inductive 
and is claimed to provide absolutely correct results. 
It is called the deductive method, and it is described 
as the method that leads from premises of general 
validity by means of logical methods of general 
validity to results of general validity. 

As a matter of fact, there is no science that does 
or could work in such a way. In the first place, we 
ask in vain, how can we arrive at such general, 
or absolutely valid, premises, since all knowledge 
is of empiric origin and is therefore equipped with 
the possibility of error as ineradicable evidence of 
this origin. In the next place, we cannot see 
how from principles at hand conclusions can be 
drawn the content of which exceeds that of 
these principles (and of the other means employed). 
In the third place, the absolute correctness of such 
results is doubtful from the fact that blunders in 
the process of reasoning cannot be excluded even 
where the premises and methods are absolutely cor 
rect. In practice it has actually come to pass that 
in the so-called deductive sciences doubts and con 
tradictions on the part of the various investigators 



Deduction 41 

of the same question are by no means excluded. To 
wit, the discussion that has been carried on for cen 
turies, and is not yet ended, over Euclid s parallel 
theorem in geometry. 

If we ask whether, in the sense of the 
observations we have just made of the formation 
of scientific principles, there is anything at all like 
deduction, we can find a procedure which bears 
a certain resemblance with that impossible pro 
cedure and which, as a matter of fact, is frequently 
and to very good purpose applied in science. It 
consists in the fact that general principles which 
have been acquired through the ordinary incom- 
1 plete induction are applied to special instances which, 
at the proposition of the principle, had not been 
taken into consideration, and whose connection with 
the general concept had not become directly evident. 
Through such application of general principles to 
cases that have not been regarded before, specific 
natural laws are obtained which had not been fore 
seen either, but which, according to the probability 
of the thesis and the correctness of the applica 
tion are also probably correct. However, the in 
vestigator, bearing in mind the factor of uncertainty 
in these ratiocinations feels in each such instance 
the need for testing the results by experience, and 
he does not consider the deduction complete until 
he had found confirmation in experience. 

Deduction, therefore, actually consists in the 
searching out of particular instances of a principle 



42 General Theory of Knowledge 

established by induction and in its confirmation by 
experience. This conduces to the growth of sci 
ence, not in breadth, but in profundity. I again 
resort to the comparison I have frequently made of 
science with a very complex network. At first 
glance we cannot obtain a complete picture of all 
the meshes. So, at the first proposition of a natural 
law an immediate survey of the entire range of the 
possible experiences to which it may apply is in- 
achievable. It is a regular, important, and neces 
sary part of all scientific work to learn the extent 
of this range and investigate the specific forms 
which the law assumes in the remoter instances. 
Now, if an especially gifted and far-seeing in 
vestigator has succeeded in stating in advance an 
especially general formulation of an inductive law, 
it is everywhere confirmed in the course of the trial 
applications, and the impression easily arises that 
confirmation is superfluous, since it results simply 
in what had already been " deduced." In point of 
fact, however, the reverse is not infrequently the 
case, that the principle is not confirmed, and condi 
tions quite different from those anticipated are 
found. Such discoveries, then, as a rule, constitute 
the starting-point of important and far-reaching 
modifications of the original formulation of the law 
in question. 

As we see, deduction is a necessary complement 
of, in fact, a part of, the inductive process. The 
history of the origin of a natural law is in general 



Deduction 43 

as follows. The investigator notices certain agree 
ments in individual instances under his observation. 
He assumes that these agreements are general, and 
propounds a temporary natural law corresponding 
to them. Then he proceeds by further experimenta 
tion to test the law in order to see whether he can 
* find full confirmation of it by a number of other 
instances. If not, he tries other formulations of 
the law applicable to the contradictory instances, or 
exclusive of them, as not allied. Through such a 
process of adjustment he finally arrives at a prin 
ciple that possesses a certain range of validity. He 
informs other scientists of the principle. These in 
their turn are impelled to test other instances known 
to them to which the principle can be applied. Any 
doubts or contradictions arising from this again im 
pel the author of the principle to carry out what 
ever readjustments may have become necessary. 
Upon the scientific imagination of the discoverer de 
pends the range of instances sufficing for the 
formulation of the general inductive principle. It 
also frequently depends upon conscious operations 
of the mind dubbed " scientific instinct." But as 
soon as the principle has been propounded, even if 
only in the consciousness of the discoverer, the de 
ductive part of the work begins, and the consequent 
test of the proposition has the most essential in 
fluence on the value of the result. 

It is immediately evident that this deductive part 
is of all the more weight, the more general the con- 



44 General Theory of Knowledge 

cepts in question are. If, in addition, the inductive 
laws posited soon prove to be of a comparatively 

1 high degree of perfection, we obtain the impression 
described above, that an unlimited number of inde 
pendent results can be deduced from a premise. 
Kant was keenly alive to the peculiarity of such a 
view, which had been widely spread pre-eminently 
by Euclid s presentation of geometry, and he gave 
expression to his opinion of it in the famous ques 
tion : How are a priori judgments possible? We 
have seen that it is not always a question of 
a priori judgments, but also of inductive conclu- 

* sions applied and tested according to deductive 

methods. 

14. Ideal Cases. Each experience may generally 
be considered under an indefinite number of various 

f concepts, all of which may be abstracted from that 
experience by corresponding observations. Accord 
ingly an indefinite number of natural laws would be 
required for prophesying that experience in all its 
parts. Likewise the indefinite number of premises 
must be known through the application of which 
those natural laws acquire a certain content. Thus 
it seems as if it were altogether impossible to apply 
natural laws for the determination of a single ex 
perience to come, and in a certain sense this is true 
(p- 3)- For example, when a child is born, we 
are quite incapable of foretelling the peculiar events 
that will occur in its life. Beyond the statement that 
it will live a while and then die, we can make only 



Ideal Cases 45 

the broadest assertions qualified by numerous " ifs " 
and " buts." 

If, in spite of this, we arrange a very great part 
of our life and activity according to the prophecies 
we make in regard to numerous details in life, bas 
ing them upon natural laws, the question arises, 
how we get over the difficulty, or, rather, the im 
possibility just referred to. 

The answer is, that we repeatedly so find or can 
form our experiences that certain natural relations 
preponderatingly determine the experience, while 
the other parts that remain undetermined fall into t 
the background. The prophecy will cover so cow- 1 
siderable a part of the experience that we can forego 
previous knowledge of the rest. We can foretell 
enough to render a practical construction of life 
possible, and increasing experience, whether .the 
personal experience of the individual or the general 
experience of science, constantly enlarges this con 
trollable part of future experiences. 

The procedure of science is similar to that of 
practical life, though freer. Whenever an in 
vestigator seeks to test a natural law of the form: 
if A is so, then B is so, he endeavors to choose 
or formulate the experiences in such a way that the 
fewest possible extraneous elements are present, and 
that those that are unavoidable should exert the 
least possible influence upon the relation in question. 
He never succeeds completely. In order, neverthe 
less, to reach a conclusion as to the form the rela- 



46 General Theory of Knowledge 

tion will take without extraneous influences, the 
following general method is applied. 

A series of instances are investigated which are 
so adjusted that the influence of the extraneous ele 
ments grows less and less. Then the relation in 
vestigated approaches a limit which is never quite 
reached, but to which it draws nearer and nearer, the 
less the influence of the extraneous elements. And 
the conclusion is drawn that if it were possible to 
exclude the extraneous elements entirely, the limit 
of the relation would be reached. 

JA case in which none of the extraneous elements 
of experience operate is called an ideal case, and the 
inference from a series of values leading to the 
limit-value is an extrapolation. Such extrapolations 
to the ideal case are a quite natural procedure in 
science, and a very large part of natural laws, espe 
cially all quantitative laws, that is, such as express 
a relation between measurable values, have precise 
validity only in ideal cases. 

We here confront the fact that many natural 
laws, and among them the most important, are ex 
pressed as, and taken to be, conditions which never 
occur in reality. This seemingly absurd procedure 
is, as a matter of fact, the best fitted for scientific 
purposes, since ideal cases are to be distinguished 
1 by this, that with them the natural laws take on the 
simplest forms. This is the result of the fact that 
in ideal cases we intentionally and arbitrarily over 
look every complication of the determining factors, 



The Determinateness of Things 47 

and in describing ideal cases we describe the simplest 
conceivable form of the class of experiences in ques 
tion. The real cases are then constructed from 
the ideal cases by representing them as the sum of 
all the elements that have an influence on the ex 
perience or the result. Just as we can represent 
the unlimited multitude of finite numbers by the 
figures up to ten, so we can represent an unlim 
ited quantity of complicated events by a finite num 
ber of natural laws, and so reach a highly service 
able approximation to reality. 

Thus geometry deals with absolutely straight 
lines, absolutely flat surfaces, and perfect spheres, 
though such have never been observed, and the re 
sults of geometry come the closer to truth, the more 
nearly the real lines, surfaces, and spheres corre 
spond to the ideal demands. Similarly, in physics, 
there are no ideal gases or mirrors, or in chemistry 
ideally pure substances, though the expressed simple 
laws in these sciences are valid for only such bodies. 
The non-ideal bodies of these sciences, which reality 
presents in various degrees of approximation, corre 
spond the more closely to these laws, the slighter the 
deviation of the real from the ideal. And the same 
method is applied in the so-called mental sciences, 
psychology and sociology, in which the " normal 
eye " and a " state with an entirely closed door " 
are examples of such idealized limit-concepts. 

15. The Determinateness of Things. A very 
widespread view and a very grave one, because of 



48 General Theory of Knowledge 

its erroneous results, is that by the natural laws 
things are unequivocally and unalterably determined 
down to the very minutest detail. This is called 
determinism, and is regarded as an inevitable con 
sequence of every natural scientific generalization. 
But an accurate investigation of actual relations 
produces something rather different. 

The most general formulation of the natural law : 
if A is experienced, then we expect B, necessarily 
refers in the first place only to certain parts of the 
thing experienced. For perfect similarity in two 
experiences is excluded by the mere fact that we 
ourselves change unceasingly and one-sidedly. Con 
sequently, no matter how accurate the repetition of 
a former experience may be, our very participation 
in it, an element bound to enter, causes it to be dif 
ferent. Therefore we deal with only a partial 
, repetition of any experience, and the common part 
is all the smaller a fraction of the entire experience, 
the more general the concept corresponding to this 
part. But the most general and most important 
natural laws apply to such very general ideas, and 
accordingly they determine only a small part of the 
whole result. Other parts are determined by other 
laws, but we can never point out an experience that 
has been determined completely and unequivocally 
by natural laws known to us. For example, we 
know that when we throw a stone, it will describe an 
approximate parabolic curv^ in falling to the 
ground. But if we should attempt to determine its 



The Determinateness of Things 49 

course accurately, we should have to take into con 
sideration the resistance of the air, the rotatory mo 
tion of the stone upon being thrown, the movement 
of the earth, and numerous other circumstances, the 
exact determination of which is a matter beyond 
the power of all sciences. Nothing but an ap 
proximate determination of the stone s course 
is possible, and every step forward toward ac 
curacy and absoluteness would require scientific 
advances which it would probably take centuries to 
accomplish. 

Science, therefore, can by no means determine 
the exact linear course that the stone will take in its 
fall. It can merely establish a certain broader path 
within which the stone s movement will remain. 
And the path is the wider the smaller the progress 
science has made in the branch in question. The 
same conditions prevail in the case of every other 
prediction based upon natural laws. Natural laws 
I merely provide a certain frame within which the 
thing will remain. But which of the infinitely nu 
merous possibilities within this frame will become 
reality can never be absolutely determined by human 
powers. 

The belief that it is possible has been evoked 
merely by a far-reaching method of abstraction on 
the part of science. By assuming in place of the 
stone " a non-extended point of mass " and by dis 
regarding all the other factors which in some way 
(whether known or unknown) exercise an influence 



50 General Theory of Knowledge 

on the stone s movement, we can effect an appar 
ently perfect solution of the problem. But the solu 
tion is not valid for real experience, merely for an 
ideal case, which bears only a more or less profound 
similarity to the real. It is only such an ideal 
world, that is, a world arbitrarily removed from its 
1 actual complexity, that has the quality of absolute 
determinateness which we are wont to ascribe to the 
real world. 

We might point to the method of abstraction gen 
erally adopted in science and to the extrapolation 
to ideal cases which has just been explained, and 
regard the assertion of the absolute determinateness 
of events in the world as a justified extrapolation to 
the ideal case. In other words, we might say that 
we know all the natural laws and how to apply 
them perfectly to the individual instances. In con- 
troversion of this it must be said that the ulterior 
justification of such ideal extrapolation is not yet 
feasible. The justification lies in the demonstra 
tion that the real cases approximate the ideal the 
more closely the more we actualize our presump 
tions. But in this case this is not feasible, since, 
for the greater part of our experiences, we do not 
even know the approximate or ideal natural laws 
by the help of which we can construct such ideal 
cases. For instance, the whole province of organic 
life is at present essentially like an unknown land, in 
which there are only a few widely separated paths 
ending in culs-de-sac. 






The Freedom of the Will 51 

1 6. The Freedom of the Will. This relation ex 
plains why, on the one hand, we assume a far- 
reaching determinateness for many things, that is, 
for all those accessible to scientific treatment and 
regulation, and why, on the other hand, we have the 

% consciousness of acting freely, that is, of being able 
to control future events according to the relations 
they bear to our wishes. Essentially there is no 
objection to be found to a fundamental deter 
minism which explains that this feeling of freedom 1 
is only a different way of saying that a part of the 
causal chain lies within onr consciousness, and that 
we feel these processes (in themselves determined) 
as if we ourselves determined their course. Nor can 
we prove this idea to be false, that, since the number 
of factors which influence each experience is in- 

* definitely great and their nature indefinitely com 
plex, each event would appear to be determined in 
the eyes of an all-comprehensive intellect. But to 
our finite minds an undetermined residue necessarily 
remains in each experience, and to that extent the 

world must always remain in part practically un 
determined to human beings. Thus, both views, 
that the world is not completely determined, and 
that it really is, though we can never recognize 
that it is, lead practically to the same result : that 
zve can and must assume in our practical atti 
tude to the world that it is only partially deter 
mined. 

But if two different lines of thought in the whole 



52 General Theory of Knowledge 

world of experience everywhere lead to the same 
result, they cannot be materially, but merely 
formally or superficially, different. For those 
things are alike which cannot be distinguished. 
There is no other definition of alikeness. Thus, if 
we see that the age-long dispute between these two 
views always breaks out afresh without seeming to 
be able to reach an end, this is readily understood, 
from what has been said, since the very same es- 
jsential arguments which can be adduced of one view 
I can be used as a prop for the other view, because in 
t their essential results the two are the same. I have 
discussed this matter because it presents a very tell- 
jing example of a method to be applied in all the 
sciences when dealing with the solution of old and 
ever recurrent moot questions. Each time we en 
counter such problems, we must ask ourselves : what 
} would be the difference empirically if the one or 
the other view were correct? In other words, we 
first assume the one to be correct, and develop the 
consequences accordingly. Then we assume the 
second to be correct and develop the consequences 
accordingly. If in the two cases the consequences 
differ in a certain definite point, we at least have 
the possibility of ascertaining the false view by in- 
t vestigating in favor of which case experience de- 
cides on this point. However, we may not con 
clude that by this the other view has been proved to 
be entirely correct. It likewise may be false, only 
with the peculiar quality that in the case in ques- 



The Classification of the Sciences 53 

tion it leads to the correct conclusions. That such a 
thing is possible, every one knows who has at 
tentively observed his own experiences. How often 
we act correctly in actual practice, though we have 
started out on false premises! The explanation 
of this possibility resides in the highly composite 
nature of each experience and each assumption. It 
is quite possible and, in fact, it is the general 
rule that a certain view contains true elements, 
but along with them false elements also. In ap 
plications of the view where the true elements are 
the decisive factors, true results are obtained, de 
spite the errors present. Likewise, false results will 
be achieved where the false elements are decisive, de 
spite the true results that can be had, or have been 
had, elsewhere, by means of the true elements. 
Hence, in case of the " confirmation," we can only 
conclude that that portion of the view essential for 
the instance in question is correct. 

One readily perceives that these observations find 
application in all provinces of science and life. There 
are no absolutely correct assertions, and even the 
falsest may in some respect be true. There are only 
greater and lesser probabilities, and every advance 
made by the human intellect tends to increase the de 
gree of probability of experiential relations, or 
natural laws. 

17. The Classification of the Sciences. From the 
preceding observations the means may be drawn 
for outlining a complete table of the sciences. How- 



54 General Theory of Knowledge 

ever, we must not regard it complete in the sense 
that it gives every possible ramification and turn of 
each science, but that it sets up a frame inside of 
which at given points each science finds its place, so 
that, in the course of progressive enlargement, the 
frame need not be exceeded. 

The basic thought upon which this classification 
rests is that of graded abstraction. We have seen 
(p. 19) that a concept is all the more general, that is, 
is applicable to all the more experiences, the fewer 
parts or elementary concepts it contains. So we 
shall begin the system of the sciences with the most 
general concepts, that is, the elementary concepts 
(or with what for the time being we shall have to 
consider elementary concepts), and, in grading the 
concept complexes according to their increasing 
diversity, set up a corresponding graded series of 
sciences. One thing more is to be noted here, that 
this graded series, on account of the very large num 
ber of new concepts entering, must produce a cor 
respondingly great number of diverse sciences. For 
practical reasons groups of such grades have been 
combined temporarily. Thereby a rougher classi 
fication, though one easier to obtain a survey of, has 
been made. The most suitable and lasting scheme 
of this sort was originated by the French philoso 
pher, August e Comte, since whom it has undergone 
a few changes. 

Below is the table of the sciences, which I shall 
then proceed to explain : 



The Classification of the Sciences 55 

I. Formal Sciences. Main concept: order 
Logic, or the science of the Manifold 
Mathematics, or the science of Quantity 
Geometry, or the science of Space 
Phoronomy, or the science of Motion 

II. Physical Sciences. Main concept : energy 

Mechanics 

Physics 

Chemistry 

III. Biological Sciences. Main concept : life 

Physiology 
Psychology 
Sociology 

As is evident, we first have to deal with the three 
great groups of the formal, the physical, and the 
biological sciences. The formal sciences treat of 
characteristics belonging to all experiences, charac 
teristics, consequently, that enter into every known 
phase of life, and so affect science in the broadest 
sense. In order immediately to overcome a wide 
spread error, I emphasize the fact that these sciences 
are to be considered just as experiential or empirical 
as the sciences of the other two groups, as to which 
there is no doubt that they are empirical. But be 
cause the concepts dealt with by the first group are 
so extremely wide, and the experiences correspond 
ing to them, therefore, are the most general of all ex 
periences, we easily forget that we are dealing with 
experiences at all; and our very firmly rooted con 
sciousness of the unqualified similarity of these ex- 



56 General Theory of Knowledge 

periences causes them to seem native qualities of the 
mind, or a priori judgments. Nevertheless, mathe- 
matics has been proved to be an empirical science 
by the fact that in certain of its branches (the 
theory of numbers) laws are known which have 
been found empirically and the " deductive " proof 
of which we have as yet not succeeded in obtaining. 
The most general concept expressed and operative in 
these sciences is the concept of order, of conjugacy 
or function, the content and significance of which 
will become clear later in a more thorough study of 
the special sciences. 

In the second group, the physical sciences, the ar 
bitrariness of the classification becomes very appar 
ent, since these sciences are among the best known. 
We are perfectly justified in regarding mechanics 
as a part of physics; and in our day physical chem 
istry, which in the last twenty years suddenly de 
veloped into an extended and important special sci 
ence, thrust itself between physics and chemistry. 

The most general concept of the physical sciences 
is that of energy, which does not appear in the 
formal sciences. To be sure it is not a fundamental 
concept. On the contrary, its characteristic is un 
doubtedly that of compositeness, or, rather, com 
plexity. 

The third group comprehends all the relations of 
living beings. Their most general concept, accord 
ingly, is that of life. By physiology is understood 
the entire science dealing with non-psychic life 



The Applied Sciences 57 

phenomena. It therefore embraces what is called, 
in the present often chance arrangement of sci 
entific activities, botany, zoology, and physiology of 
the plants, animals, and man. Psychology is the 
science of mental phenomena. As such, it is not 
limited to man, even though for many reasons he 
claims by far the preponderating part of it for him 
self. Sociology is the science which deals with the 
peculiarities of the human race. It may therefore 
be called anthropology, but in a far wider sense than 
the word is now applied. 

18. The Applied Sciences. It will be remarked ) 
that the grouping of the table gives no place at all ! 
in its scheme to certain branches of learning taught 
in the universities and equally good technical in 
stitutions. We look in vain not only for theology : 
and jurisprudence, but also for astronomy, medi- ) 
cine, etc. 

The explanation and justification of this is, that 
for purposes of systematization we must distinguish 
between pure and applied sciences. By virtue of 
their strictly conceptual exclusiveness the pure sci 
ences constitute a regular hierarchy or graded series, 
so that all the concepts that have been used and dealt 
with in the preceding sciences are repeated in the fol 
lowing sciences, while certain characteristic new con 
cepts enter in addition. Thus logic, the science of 
the manifold, exercises its dominion over all the 
other sciences, while the specific concepts of physics 
and chemistry have nothing to do with it, though 



58 General Theory of Knowledge 

they are of importance to all the biologic sciences. 
Through this graded addition of new (naturally em 
piric) concepts, the construction of the pure sciences 
proceeds in strict regularity, and their problems arise 
exclusively from the application of new concepts to 
all the earlier ones. In other words, their problems 
do not reach them accidentally from without, but re 
sult from the action and reaction of their concepts 
upon one another. 

At the same time there are problems that each day 
sets before us without regard to system. These 
come from our endeavor to improve life and avert 
evil. In the problems of life we are confronted 
by the whole variety of possible concepts, and 
under the day s immediate compulsion we can 
not wait, if we are sowing crops or helping a sick 
man, until physiology and all the other appropriate 
sciences have solved all the problems of plant 
growth and the changes of the human body and 
human energy. When other signs fail, we use the 
position of the stars for finding our way on the 
high seas. In this manner we turn the teaching of 
the stars, or astronomy, into an applied science, in 
which at first mechanics alone seemed to have a part. 
Later physics took a share in it, then optics took a 
particularly prominent share, and in recent times 
not only did chemistry find its way into astronomy, 
but the specifically biologic concept of evolution was 
applied in astronomy with success. 

Thus, side by side with the pure sciences are the 



The Applied Sciences 59 

applied, which are to be distinguished from the pure 
sciences by the fact that they do not unfold their 
problems systematically, but are assigned them by 
the external circumstances of man s life. The pure 
sciences, therefore, almost always have a larger or 
smaller share in the tasks of the applied sciences. 
For instance, in building a bridge or railroad, phys 
ical problems have to be taken into consideration as 
well as sociologic problems (problems of trade), 
and a good physician should be a psychologist as 
well as a chemist. 

But since all the individual questions arising in 
the applied sciences may be considered essentially as 
problems of one or other pure science, they need 
not be explicitly enumerated along with the pure sci 
ences, especially since their development is greatly 
dependent upon temporary conditions and is there 
fore incapable of simple systematization. 



PART II 

LOGIC, THE SCIENCE OF THE MANIFOLD, 
AND MATHEMATICS 

19. The Most General Concept. If we try to con 
ceive the whole structure of science according to 
the principle of the increasing complexity of con 
cepts, the first question which confronts us is, What- 
concept is the most general of all possible concepts, 
so general that it enters into every concept forma 
tion and acts as a decisive factor ? In order to find 
this concept let us go back to the psycho-physical 
basis of concept formation, namely, memory, and 
let us investigate what is the general characteristic 
determining memory. We soon perceive that if a 
being were to lead an absolutely uniform existence, 
no memories could be evoked. There would be 
nothing by which the past could be distinguished 
from the present, hence nothing by which to com 
pare them. So the " primal phenomenon "of con 
scious thought is the realization of a difference, a 
difference between memory and the present, or, to 
put the same idea still more generally, between two 
memories. 

Our experiences, therefore, are divided into two 

61 



62 Logic and Mathematics 

parts, distinguished from each other. In order to 
predicate something of a perfectly general nature 
concerning those parts, without regard to their par 
ticular content, we must, in accordance with the 
means employed in human intercourse, designate 
them by a name. Now in all human languages there 
is a great deal of arbitrariness and indefiniteness in 
the relations between the concepts and the names ap 
plied to them, which render all accurate work in the 
study of concepts extremely difficult. It is neces 
sary, therefore, to state definitely in each particular 
instance with what conceptual content a given name 
is to be connected. Every experience in so far as 
it is differentiated from other experiences we shall 
call simply an experience without making a distinc 
tion between a so-called inner or outer experience. 

Many of the experiences remain isolated, because 
they are not repeated in a similar form, and so do 
not remain in our memory. They depart from our 
psychic life once for all and leave no further conse 
quences or associations. But some experiences re 
cur with greater or less uniformity, and become per 
manent parts of psychic life. Their duration is by 
no means unlimited. For even memories fade and 
disappear. However, they extend through a con 
siderable part of life, and that suffices to give them 
their character. 

The aggregate of similar experiences, hence of 
experiences conceptually generalized, we shall call 
things. A thing, therefore, is an experience which 



Association 63 

has been repeated, and is " recognized " by us. 
That is, it is felt as repeated and conceptually com 
prehended. In other words, all experiences of 
which we have formed concepts are things, and the 
concept of thing itself is the most general concept, 
since, according to its definition, it includes all pos 
sible concepts. Its " essence," or determining char 
acteristic, lies in the possibility of differentiating any 
one thing from another. Things we do not differ 
entiate we call the same, or identical. Here we 
shall leave undecided the question whether this lack 
of differentiation occurs because we cannot, or be 
cause we would not, differentiate. All experiences 
generalized into one concept are therefore felt or re 
garded as the same in reference to this concept. 
Now, since concepts arise unconsciously as well as 
consciously, the first is a case of identities which 
had been directly felt as such. On the other hand, 
in the second case, the process is that of consciously 
disregarding or abstracting the existing differences 
in order to form a concept into which these do not 
enter. This last process is applied in the highest 
degree possible in obtaining the concept thing. 

20. Association. The experience of the connec 
tion or relation between various things is also de 
rived from the nature of our experiences in the most 
general sense. When we recall a thing A, an 
other thing B comes to our mind, the memory of 
which is called forth by A, and vice versa. The 
cause of this invariably lies in some experiences in 



64 Logic and Mathematics 

which A and B occur together. In fact, A and B 
must have occurred together a number of times. 
Otherwise they would have disappeared from mem 
ory. In other words, it is the fact of the complex 
concept which appears in such connections between 
various things. Two things, A and B, which are con 
nected with each other in such a way, are said to be 
associated. Association in the most general sense 
means nothing more than that when we think of B 
we also have A in our consciousness, and vice versa. 
However, we can at will make the association more 
definite, so that quite definite thoughts or actions 
will be connected with the association of B. These 
thoughts and actions are then the same for all the 
individual cases occurring under the concept A 
and B. 

If we associate with the thing B another thing 
C, we obtain a relation of the same nature as that 
obtained by the association of A and B. But at the 
same time a new relation arises which was not di 
rectly sought, namely, the association of A to C. 
If A recalls B, and B recalls C, A must inevitably 
recall C also. This psychologic law of nature is 
productive of numberless special results. For we 
can apply it directly to still another case, the as 
sociation of a fourth thing D to the thing C, whereby 
new relations are necessarily established also be 
tween A and D as well as between B and D. By 
positing the one relation C : D there arise two new 
relations not immediately given, namely, A : D and 



The Group 65 

B : D. The reason the other relations arise is be 
cause C was not taken free from all relations, but 
had already attached to it the relations to A and B. 
These relations of C, therefore, brought A and B 
into the new relation with D. 

By this simplest and most general example we 
recognize the type of the deductive process (p. 41), 
namely, the discovery of relations which, it is true, 
have already been established by the accepted 
premises, but which do not directly appear in under 
taking the corresponding operations. In the pres 
ent case, to be sure, the deduction is so apparent 
that the recognition of the relations in question of 
fers not the slightest difficulty. But we can easily 
imagine more complicated cases in which it is much 
more difficult to find the actually existing relations, 
and so in certain circumstances we may search for 
them a long time in vain. 

21. The Group. The aggregate of all. individual 

wHte 

things occurring in a definite concept, or^the com 
mon characteristics of which make up this concept, 
is called a group. Such a group may consist of a 
limited or finite number of members, or may be 
unlimited, according to the nature of the concepts 
that characterize it. Thus, all the integers form an 
unlimited or infinite group, while the integers be 
tween ten and one hundred (or the two-digit num 
bers) form a limited or finite group. 

From the definition of the group concept follows 
the so-called classic process of argumentation of the 



66 Logic and Mathematics 

syllogism. Its form is : Group A is distinguished by 
the characteristic of B. The thing C belongs to 
group A. Therefore C has the characteristic of B. 
The prominent part ascribed by Aristotle and his 
successors to this process is based upon the certainty 
which its results possess. Nevertheless, it has been 
pointed out, especially by Kant, that judgments or 
conclusions of such a nature (which he called ana 
lytic) have no significance at all for the progress 
of science, since they express only what is already 
known. For in order to enable us to say that the 
thing C belongs to group A, we must already have 
recognized or proved the presence of the group 
characteristic B in C, and in that case the conclu 
sion only repeats what is already contained in the 
second or minor premise. 

This is evident in the classic example : All men 
are mortal. Caius is a man. Therefore Caius is 
mortal. For if Caius s mortality were not known 
(here we are not concerned how this knowledge was 
obtained), we should have no right to call him a 
man. 

At the same time the character of the really sci 
entific conclusion based upon the incomplete induc 
tion becomes clear. It proceeds according to the 
following form. The attributes oLthe group A are 
the characteristics of a, b, c, d. I We find in the 
thing C the characteristics a, b, c. Therefore we 
presume that the characteristic d will also be found 
in C. The ground for this presumption is that we 



The Group 67 

have learned by experience that the characteristics 
mentioned have always been found together. It is 
for this reason, and for this reason only, that we 
may assume from the presence of a, b, c the pres 
ence of d. In the case of an arbitrary combination, 
in which it is possible to combine other character 
istics, the conclusion is unfounded. But if, on the 
other hand, the formation of the concept A with the 
characteristics of a, b, c, d has been caused by re 
peated and habitual experience, then the conclusion 
is well founded; that is, it is probable. 

As a matter of fact, however, that classic ex 
ample which is supposed to prove the absolute cer 
tainty of the regular syllogism turns out to be a hid 
den inductive conclusion of the incomplete kind. 
The premise, Caius is a man, is based on the at 
tributes a, b, c (for example, erect bearing, figure, 
language), while the attribute d (mortality) cannot 
be brought under observation so long as Caius re 
mains alive. In the sense of the classic logic, there 
fore, we are not justified in the minor premise, 
Caius is a man, while Caius is alive. The utter 
futility of the syllogism is apparent, since, accord 
ing to it, it is only of dead men that we can assert 
that they are mortal. 

From these observations it becomes further appar 
ent that logic, whether it is the superfluous classic 
logic or modern effective inductive logic, is nothing 
but a part of the group theory, or science of mani- 
foldness, which appears as the first, because it is 



68 Logic and Mathematics 

the most general member of the mathematical sci 
ences (this word taken in its widest significance). 
But according to the hierarchic system in harmony 
with which the scheme of all the sciences had been 
consciously projected, we cannot expect anything 
else than that those sciences which are needful 
for the pursuit of all other sciences (and logic has 
always been regarded as such an indispensable sci 
ence, or, at least, art) should be found collected and 
classified in the first science. 

22. Negation. When the characteristics a, b, c, d 
of a group have been determined, then the aggregate 
of all things existing can be divided into two parts, 
namely, the things which belong to the group A and 
those which do not belong to it. This second ag 
gregate may then be regarded as a group by itself. 
If we call this group " not-A," it follows from 
the definition of this group that the two groups, 
A and not-A, together form the aggregate of all 
things. 

This is the meaning and the significance of the 
linguistic form of negation. It excludes the thing 
negated from any group given in a proposition, and 
this relegates it to the second or complementary 
group. 

The characteristic of such a group is the common 
absence of the characteristics of the positive group. 
We must note here that the absence of even one of 
the characteristics a, b, c, d excludes the incorpora 
tion of the thing into the group A, while the mere 



Artificial and Natural Groups 69 

absence of this characteristic suffices to include it 
in the group not-A. We can therefore by no means 
predicate of group not-A that each one of its mem 
bers must lack all the characteristics a, b, c, d. We 
can only say that each of its members lacks at least 
one of the characteristics, but that one or some may 
be present, and several or all may be absent. From 
this follows a certain asymmetry of the two groups, 
which we must bear in mind. 

The consideration of this subject is especially im 
portant in the treatment of negation in the conclu 
sions of formal logic. As we shall make no special 
use of formal logic, we need not enter into it in 
detail. 

23. Artificial and Natural Groups. The combina 
tion of the characteristics which are to serve for the 
definition of a group is at first purely arbitrary. 
Thus, when we have chosen such an arbitrary com 
bination, a, b, c, d, we can eliminate one of the 
characteristics, as, for example, c, and form a group 
with the characteristics a, b, d. Such a group, 
which is poorer in characteristics, will, in general, be 
richer in members, for to it belong, in the first 
place, all the things with the characteristics a, b, c, d, 
of which the first group consisted, and in addition 
all the things which, though not possessing c, pos 
sess a, b, and d. 

If we call such groups related as contain common 
characteristics, though containing them in different 
members and combinations, so that the definition of 



70 Logic and Mathematics 

the one group can be derived from the other by the 
elimination or incorporation of individual charac 
teristics, then we can postulate the general thesis 
that in related groups those must be richer in mem 
bers which are poorer in characteristics, and in 
versely. This is the precise statement of the 
proposition of the less definite thesis stated above. 

For the purposes of systematization we have as 
sumed that we can arbitrarily eliminate one or an 
other characteristic of a group. In experience, 
however, this often proves inadmissible. As a rule 
we find that the things which lack one of the char 
acteristics of a group will also lack a number of 
other characteristics ; in other words, that the char 
acteristics are not all independent of one another, 
but that a certain number of them go together, so 
that they are present in a thing either in common 
or not at all. 

This case, however, can be referred to the gen 
eral one first described, by treating the character 
istics belonging together as being one characteristic, 
so that the group is defined solely by the independ 
ent characteristics. Then, according to the defini 
tion, we can, without losing our connection with ex 
perience, carry out that formal manifoldness of all 
possible related groups which yields what is called 
a classification of the corresponding things. 

If for the determination of a group a definite 
number of independent characteristics is taken, say, 
a, b, c, d, and e, then we have at first the narrowest 



Artificial and Natural Groups 71 

or poorest group abcde. By the elimination of one 
characteristic we obtain the five groups, bcde, acde, 
abde, abce, and abed. If we omit one other char 
acteristic we get ten different groups abc, abd, abe, 
acd, ace, ade, bed, bee, bde, cde. Likewise, there 
are ten groups with two characteristics each, and 
finally five groups with one characteristic each. All 
these groups are related. There is a science, the 
Theory of Combinations, which gives the rules by 
which, in given elements or characteristics, the kind 
and number of the possible groups can be found. 
The theory of combinations enables us to obtain a 
complete table and survey of all possible complex 
concepts which can be formed from given simple 
ones (whether they be really elementary concepts, 
or only relatively so) . When in any field of science 
the fundamental concepts have been combined in 
this manner, a complete survey can be had of all the 
possible parts of this science by means of the theory 
of combinations. 

In order to present this process vividly to our 
minds, let us take as an example the science of the 
chemical combination of substances which form an 
important part of chemistry. There are about 
eighty elements in chemistry, and this science has 
to treat of 

a) each of the eighty elements by itself 

b) all substances containing two elements and no 

more 

c) all substances containing three elements 



72 Logic and Mathematics 

d, e, f, etc.) the substances containing four, five, 

and six, etc., elements, 

until finally we reach a group (not existing in ex 
perience) embracing substances formed of all the 
elements. That there is no such substance in the 
present scope of human knowledge has, of course, 
no significance for the structure of the scheme. 
What is significant is the fact that the scheme really 
embraces and arranges all possible substances in 
such a way that we cannot conceive of any case in 
which a newly discovered substance cannot after 
examination immediately be classed with one of the 
existing groups. 

To cite an example from another science. 
Physics, it will be recalled, may be considered to be 
the science of the different kinds of energy. This 
science, accordingly, is divided first into the study 
of the properties of each energy, and then into the 
study of the relations of two energies, of three 
energies, of four energies, etc. Here, too, we may 
say that in the end there can be no physical phenom 
enon which cannot be placed in one of the groups so 
obtained. 

Of course, neither in chemistry nor in physics 
does this mean that each new case will fall within 
the scheme obtained by the exhaustive combination 
of elementary concepts (whether chemical elements 
or kinds of energy) known at the time. It is quite 
possible that a new r thing under investigation con 
tains a nciv elementary concept, so that on account 



Artificial and Natural Groups 73 

of it the scheme must be enlarged through the em 
bodiment of this new element. But simultaneously 
a corresponding number of new groups appear in 
the scheme, and the investigator s attention is di 
rected to the fact that he still has a reasonable pros 
pect, in favorable circumstances, of discovering 
these new things also. Thus combinatory schemati- 
zation serves not only to bring the existing content 
of science into such order that each single thing has 
its assigned place, but the groups which have thereby 
been found to be vacant, to which as yet nothing of 
experience corresponds, also point to the places in 
which science can be completed by new discoveries. 

From the above presentation it is apparent how 
from the two concepts " thing " and " association " 
alone a great manifoldness of various and regular 
forms can be developed. They are purely empirical 
relations, for the fact that several things can be 
combined in the graded series described above ac 
cording to a fixed rule does not follow merely from 
the two concepts, but must be experienced. But, on 
the other hand, both concepts are so general that the 
experiences obtained in some cases can be applied to 
all possible experiences and may serve the purpose 
of classifying and making a general survey of 
them. 

The above statements, however, have by no means 
exhausted the possibilities. For it has been tacitly 
assumed that in the combination of several things 
the sequence according to which this combination 



74 Logic and Mathematics 

takes place should not condition a difference of the 
result. This is true of a number of things, but not 
of all. In order, therefore, to exhaust the possi 
bilities the theory of combinations must be extended 
also to cases in which the sequence is to be taken 
account of, so that the form ab is regarded as differ 
ent from ba. 

We will not undertake to work out the results of 
this assumption. It is obvious that the manifold- 
ness of the various cases is much greater than if we 
neglect the sequence. On this point we have one 
more observation to make, that further causes for 
diversity exist. It is true that a chemical combina 
tion is not influenced by the sequence in which its 
elements enter the combination, but there do occur 
with the same elements differences in their quanti 
tative relations, and thereby a new complexity is in 
troduced into the system, so that two or more 
similar elements can form different combinations 
according to the difference in the quantitative rela 
tions. Still, even with this, the actual manifold- 
ness is not exhausted, for from the same elements 
and with the same quantitative relations there can 
arise different substances called isomeric, which, for 
all their similarity, possess different energy con 
tents. But the first scheme is not demolished, nor 
does it become impracticable because of this increase 
of manifoldness. What simply happens is that sev 
eral different things instead of one appear in the 
same group of the original scheme, the systematic 



Arrangement of the Members 75 

classification of which necessitates a further 
schematization by the use of other characteristics. 

24. Arrangement of the Members. Since we 
have started from the proposition that all members 
of a group are different from one another, we have 
perfect liberty to arrange them. The most obvious 
arrangement according to which some one definite 
member is followed by a single other member and 
so forth (as, for example, the arrangement of the 
letters of the alphabet) is by no means the only mode 
of arrangement, though it is the simplest. Besides 
this linear arrangement, there is also, for instance, 
the one in which two new members follow simul 
taneously upon each previous one, or the members 
may be disposed like a number of balls heaped up 
in a pyramid. However, we shall not have much 
occasion to occupy ourselves with these complex 
types of arrangement, and can therefore limit our 
considerations at first to the simplest, that is, to the 
linear arrangement. 

This simplest of all possible forms expresses itself 
in the fact that the immediately experienced things 
of our consciousness are arranged in this way. In 
point of fact, the contents of our consciousness pro 
ceed in linear order, one single new member always 
attaching itself to an existing member. This law, 
however, is not strictly and invariably adhered to. 
It sometimes happens that our consciousness con 
tinues for a while to pursue the direction of thought 
it has once taken, although a branching off had al- 



76 Logic and Mathematics 

ready taken place at a former point, at which a new 
chain of thought had begun. Nevertheless, one of 
these chains usually breaks off very soon, and the 
linear character of the inner experience is immedi 
ately restored. Of certain specially powerful in 
tellects it is recorded that they could keep up several 
lines of thought for a considerable length of time 
Julius Caesar, for instance. 

The biologic peculiarity here mentioned of the 
linear juxtaposition of the contents of our con 
sciousness has led to the concept of time, which has 
been appropriately called a form of inner life. That 
all our experiences succeed each other in time is 
equivalent to saying that our thought processes rep 
resent a group in linear arrangement. As appears 
from the above observations, this is by no means an 
absolute form, unalterable for all times. On the 
contrary, a few highly developed individuals have 
already begun to emancipate themselves from it. 
But the existing form is so firmly fixed through 
heredity and habit that it still seems impracticable 
for most men to imagine the succession of the inner 
experiences in a different way than by a line or by 
one dimension. Since, on the other hand, we have 
all learned to feel space as in-dimensional, although 
optically it appears to possess only two dimensions 
(we see length and breadth, and only infer thick 
ness from secondary characteristics), we come to 
recognize that the linear form by which we repre 
sent the succession of our experiences is a matter 



Arrangement of the Members 77 

of adaptation, and that because the change has been 
extremely slight in the course of centuries it pro 
duces the impression of being unalterable.* 

These discussions lead to a further difference 
that can exist in groups of linear arrangement. 
While in the first example we chose, the alphabet, 
the sequence was quite arbitrary, since any other 
sequence is just as possible, the same cannot be said 
of experiences into which the element of time en 
ters. These are not arbitrary, but are arranged by 
special circumstances depending upon the aggregate 
of things which co-operate in the given experi 
ences. 

While, therefore, a group with free members, 
that is, members not determined in their arrange 
ment by special circumstances, can be brought into 
linear order in very different ways, there are groups 
in which only one of those orders actually occurs. 
We see at once that in free groups the number of 
different orders possible is the greater, the greater 
the group itself. The theory of combinations 
teaches how to calculate these numbers which play 
a very important role in the various provinces of 

* Mathematicians who busy themselves a great deal with the 
formal theory of four-dimensional space, seem to acquire a 
capacity for imagining this form as easily as the three-dimen 
sional form with which we are all familiar. Therefore, despite 
the oft-repeated statements to the contrary, it is not impossible 
to imagine four-dimensional space. Only, we must not at 
tempt to represent to ourselves four-dimensional space in 
three-dimensional space, especially not without a knowledge 
of its properties, 



78 Logic and Mathematics 

mathematics. The naturally ordered groups always 
represent a single instance out of these possibilities, 
the source of which always lies outside the group 
concept, that is, it proceeds from the things them 
selves which are united into a group. 

25. Numbers. An especially important group in 
the linear order is that of the integral numbers. Its 
origin is as follows: 

First we abstract the difference of the things 
found in the group, that is, we determine, although 
they are different, to disregard their differences. 
Then we begin with some member of the group and 
form it into a group by itself. It does not matter 
which member is chosen, since all are regarded as 
equivalent. Then another member is added, and 
the group thus obtained is again characterized as a 
special type. Then one more member is added, 
and the corresponding type formed, and so on. Ex 
perience teaches that never has a hindrance arisen 
to the formation of new types of this kind by the 
addition of a single member at a time, so that the 
operation of this peculiar group formation may be 
regarded as unlimited or infinite. 

The groups or types thus obtained are called the 
integral numbers. From the description of the 
process it follows that every number has two neigh 
bors, the one the number from which it arose by 
the addition of a member, and the other the number 
which arose from it by the addition of a member. 
In the case of the number one with which the series 



Arithmetic 79 

begins, this characteristic is present in a peculiar 
form, the preceding group being group zero, that is, 
a group without content. This number in conse 
quence reveals certain peculiarities into which we 
cannot enter here. 

Now, according to a previous observation (p. 
64), not only does the order bring every number 
into relation with the preceding one, but since this 
last for its part already possesses a great number 
of relations to all preceding, these relations exert 
their influence also upon the new relation. This 
fact gives rise to extraordinarily manifold relations 
between the various numbers and to manifold laws 
governing these relations. The elucidation of them 
forms the subject of an extensive science. 

26. Arithmetic, Algebra, and the Theory of Num 
bers. From this regular form of the number series 
numerous special characteristics can be established. 
The investigations leading to the discovery of these 
characteristics are purely scientific, that is, they have 
no special technical aim. But they have the un 
commonly great practical significance that they pro 
vide for all possible arrangements and divisions of 
numbered things, and so have instruments at hand 
ready for application to each special case as it 
arises. I have already pointed out that in this 
lies the positive importance of the theoretical sci 
ences. For practical reasons the study of them 
must be as general as possible. This science is 
called arithmetic. 



80 Logic and Mathematics 

Arithmetic undergoes an important generaliza 
tion if the individual numbers in a calculation are 
disregarded and abstract signs standing for any 
number at all are used in their place. At first 
glance this seems superfluous, since in every real 
numerical calculation the numbers must be reintro- 
duced. The advantage lies in this, that in calcula 
tions of the same form, the required steps are for 
mally disposed of once for all, so that the numerical 
values need be introduced only at the conclusion and 
need not be calculated at each step. Moreover, the 
general laws of numerical combination appear much 
more clearly if the signs are kept, since the result 
is immediately seen to be composed of the participat 
ing members. Thus, algebra, that is, calculation 
with abstract or general quantities, has developed as 
an extensive and important field of general mathe 
matics. 

By the theory of numbers we understand the most 
general part of arithmetic which treats of the prop 
erties of the " numerical bodies " formed in some 
regular way. 

27. Co-ordination. So far our discussion has 
confined itself to the individual groups and to the 
properties which each one of them exhibits by itself. 
We shall now investigate the relations which exist 
between two or more groups, both with regard to 
their several members and to their aggregate. 

If at first we have two groups the members of 
which are all differentiated from one another, then 



Co-ordination 81 

any one member of the one group can be co-ordin 
ated with any one member of the other group. This 
means that we determine that the same should be 
done with every member of the second group as is 
done with the corresponding member of the first 
group. That such a rule may be carried out we 
must be able to do with the members of all the 
groups whatever we do with the members of one 
group. In other words, no properties peculiar to 
individual members may be utilized, but only the 
properties that each member possesses as a member 
of a group. As we have seen, these are the prop 
erties of association. 

First, the co-ordination is mutual, that is, it is im 
material to which of the two groups the processes 
are applied. The relation of the two groups is 
reciprocal or symmetrical. 

Further, the process of co-ordination can be ex 
tended to a third and a fourth group and so on, 
with the result that what has been done in one of 
the co-ordinated groups must happen in all. If 
hereby the third group is co-ordinated with the 
second, the effects are quite the same as if it were 
co-ordinated directly with the first instead of in 
directly through the second. And the same is true 
for the fourth and the fifth groups, etc. Thus, co 
ordination can be extended to any number of groups 
we please, and each single group proves to be co 
ordinated with every other. 

Finally, a group can be co-ordinated with itself, 



82 Logic and Mathematics 

each of its members corresponding to a certain 
definite other member. It is not impossible that 
individual members should correspond to them 
selves, in which case the group has double members, 
or double points. The limit-case is identity, in 
which every member corresponds to itself. This 
last case cannot supply any special knowledge in it 
self, but may be applied profitably to throw light 
on those observations for which it represents the ex 
treme possibility. 

28. Comparison. If we have two groups A and 
B, and if we co-ordinate their members severally, 
three cases may arise. Either group A is exhausted 
while there are members remaining in B, or B is 
exhausted before A, or, finally, both groups allow 
of a mutual co-ordination of all their members. In 
the first case A is called, in the broader sense of the 
word, smaller than B, in the second B is called 
smaller than A, in the third the two groups are said 
to be of equal magnitude. The expression, " B is 
greater than A," is equivalent to the expression, " A 
is smaller than B," and inversely. 

It is to be noted that the relations mentioned 
above are true, whether the members are considered 
as individually different from one another or 
whether the difference of the members is disre 
garded, and they are treated as alike. This comes 
from the fact that every definite co-ordination of a 
group can be translated into every other possible co 
ordination by exchanging two members at a time in 



Comparison 83 

pairs. Since in this process one member is each 
time substituted for another, and a gap therefore can 
never occur in its place, the group in the new ar 
rangement can be co-ordinated with the other group 
as successfully as in the old arrangement. At the 
same time we learn from this that in every co-ordi 
nation of a group with itself, independently of the 
arrangement of its members, it must prove equal to 
itself. 

By carrying out the co-ordination proof is further 
supplied of the following propositions : 

( greater than ^ 

If group A is -< equal to I group B 
( smaller than J 

f greater than \ 

and group B is < equal to v group C 
( smaller than ) 

( greater than ") 

then group A is also J equal to v group C 

( smaller than j 

From this it follows that any collection of finite 
groups whatsoever, of which no one is equal to the 
other, can always be so arranged that the series 
should begin with the smallest and end with the 
greatest, and that a larger should always follow a 
smaller. This order would be unequivocal, that is, 



84 Logic and Mathematics 

there is only one series of the given groups which 
has this peculiarity. As we shall soon see, the 
series of integers is the purest type of a series so ar 
ranged. 

In comparing two infinitely large groups by co 
ordination, it may be said on the one hand that never 
will one group be exhausted while the other still 
contains members. Accordingly, it is possible to 
designate two unlimited or infinite groups (or as 
many such groups as we please) as equal to each 
other. On the other hand, the statement that in 
both groups each member of the one is co-ordinated 
with a member of the other has no definite meaning 
on account of the infinitely large number of mem 
bers. The definition of equality is therefore not 
completely fulfilled, and we must not loosely apply 
a principle valid for finite groups to infinite groups. 
This consideration, which may assume very dif 
ferent forms according to circumstances, explains 
the " paradoxes of the infinite," that is, the con 
tradictions which arise when concepts of a definite 
content are applied to cases possessing in part a dif 
ferent content. If we wish to attempt such an ap 
plication, we must in each instance make a special 
investigation as to the manner in which the rela 
tions on their part change by the change of those 
contents (or premises). As a general rule we must 
expect that the former relations will not remain 
valid in these circumstances without any change at 
all. 



Counting 85 

In the course of these observations we have 
learned how co-ordination can be used for obtain 
ing a number of fundamental and multifariously 
applied principles. From this alone the great im 
portance of co-ordination is evident, and later we 
shall see that its significance is even more far- 
reaching. The entire methodology of all the sci 
ences is based upon the most manifold and many- 
sided application of the process of co-ordination, 
and we shall have occasion to make use of it re 
peatedly. Its significance may be briefly charac 
terized by stating that it is the most general means 
of bringing connection into the aggregate of our 
experiences. 

29. Counting. The group of integral numbers, 
because of its fundamental simplicity and regular 
ity, is by far the best basis of co-ordination. For 
while arithmetic and the theory of numbers give 
us a most thorough acquaintance with the peculi 
arities of this group, we secure by the process of co 
ordination the right to presuppose these peculiarities 
and the possibility of finding them again in every 
other group which we have co-ordinated with the 
numerical group. The carrying out of such co 
ordination is called counting, and from the premises 
made it follows that we can count all things in so 
far as we disregard their differences. 

We count when we co-ordinate in turn one mem 
ber of a group after another with the members of 
the number series that succeed one another until 



86 Logic and Mathematics 

the group to be counted is exhausted. The last 
number required for the co-ordination is called the 
sum of the members of the counted group. Since 
the number series continues indefinitely, every given 
group can be counted. 

Numerals have been co-ordinated with iiames as 
well as with signs. The former are different in 
the different languages, the latter are international, 
that is, they have the same form in all languages. 
From this proceeds the remarkable fact that the 
written numbers are understood by all educated 
men, while the spoken numbers are intelligible only 
within the various languages. 

The purpose of counting is extremely manifold. 
Its most frequent and most important application 
lies in the fact that the amount affords a measure for 
the effectiveness or the value of the corresponding 
group, both increasing and decreasing simul 
taneously. A further number serves as a basis for 
divisions and arrangements of all kinds to be car 
ried out within the group, whereby liberal use is 
made of the principle that everything that can be 
effected in the given number group can also be ef 
fected in the co-ordinated counted group. 

30. Signs and Names. The co-ordination of 
names and signs with numbers calls for a few gen 
eral remarks on co-ordination of this nature. 

The possibility of carrying out the formal opera 
tions effected in one of the groups upon the co 
ordinated group itself facilitates to an extraordinary 



Signs and Names 87 

extent the practical shaping of the reality for 
definite purposes. If by counting we have ascer 
tained that a group of people numbers sixty, we 
can infer without actually executing the steps that 
it is possible to form these men in six rows of ten, 
or in five rows of twelve, or in four rows of fifteen, 
but that we cannot obtain complete rows if we try 
to arrange them in sevens or elevens. These and 
numberless other peculiarities we can learn of tne 
group of men from its amount, that is, from its co 
ordination with the numerical group of sixty. In 
co-ordination, therefore, we have a means of ac 
quainting ourselves with facts without having to 
deal directly with the corresponding realities. 

It is clear that men will very soon notice and 
avail themselves of so enormous an advantage for 
the mastery and shaping of life. Thus, we see the 
process of co-ordination in general use among the 
most primitive men. Even the higher animals 
know how to utilize co-ordination consciously. 
When the dog learns to answer to his name, when 
the horse responds to the " Whoa " and the " Gee " 
of his driver there is in each case a co-ordination 
of a definite action or series of actions, that is, of a 
concept with a sign, or, in other words, of a con 
cept with a member of another group; and in this 
there need not be the least similarity between the 
things co-ordinated with each other. The only re 
quirement is that on the one hand the co-ordinated 
sign should be easily and definitely expressed and be 



88 Logic and Mathematics 

to the point, and that, on the other hand, it should be 
easily " understood," that is, comprehended by the 
senses and unmistakably differentiated from other 
signs co-ordinated with other things. 

Thus, we find that the most frequent concepts of 
co-ordinated sound signs form the beginnings of 
language in the narrower sense. It is very difficult 
to ascertain for what reasons the particular forms 
of sound signs have been chosen, nor is it a matter 
of great importance. In the course of time the 
original causes have disappeared from our con 
sciousness and the present connection is purely ex 
ternal. This is evident from the enormous differ 
ence of languages in which hundreds of different 
signs are employed for the same concept. 

Now it would be quite possible to solve the prob 
lem of co-ordinating with each group of concepts a 
corresponding group of sounds, so that each con 
cept should have its own sound, or, in other words, 
that the co-ordination should be unambiguous. It 
would not by any means be beyond human power to 
accomplish this, if it were not for the fact that the 
concepts themselves are still in so chaotic a state 
as they are at present. We have seen that the at 
tempts of Leibnitz and Locke to draw up a system 
of concepts, if only in broad outline, have undergone 
no further development since. Even the most reg 
ulated concepts as well as the familiar concepts of 
daily life are in ceaseless flux, while the co-ordinated 
signs are comparatively more stable. But they, 



The Written Language 89 

too, undergo a slow change, as the history of lan 
guages shows, and in accordance with quite different 
laws from those which govern the change of con 
cepts. The consequence is that in language the co 
ordination of concepts and words is far from being 
unambiguous. The science of language designates 
the presence of several names for the same con 
cept and of several concepts for the same name by 
the words synonym and homonym. These forms, 
which have arisen accidentally, signify so many 
fundamental defects of language, since they destroy 
the principle of unambiguity upon which language 
is based. In consequence of the false conception 
of its nature we have until now positively shrunk 
from consciously developing language in such a way 
that it should more and more approach the ideal of 
unambiguity. Such an ideal is in fact scarcely 
known, much less recognized. 

31. The Written Language. Sound signs, to be 
sure, possess the advantage of being produced easily 
and without any apparatus, and of being communi 
cable over a not inconsiderable distance. But they 
suffer under the disadvantage of transitoriness. 
They suffice for the purpose of temporary under 
standing and are constantly being used for that. 
If, on the other hand, it is necessary to make com 
munications over greater distances or longer periods 
of time, sound signs must be replaced by more per 
manent forms. 

For this we turn to another sense, the sense of 



90 Logic and Mathematics 

sight. Since optic signs can travel much greater 
distances than sound signs without becoming indis 
tinguishable, we first have the optical telegraphs, 
which find application, though rather limited ap 
plication, in very varying forms, the most efficient 
being the heliotrope. The other sort of optic signs 
is much more generally used. These are objectively 
put on appropriate solid bodies, and last and are un 
derstood as long as the object in question lasts. 
Such signs form the written language in the widest 
sense, and here, too, it is a question of co-ordinating 
signs and concepts. 

What I have said concerning the very imperfect 
state of our present concept system is true also of 
these two groups. On the other hand, the written 
signs are not subject to such great change as the 
sound signs, because the sound signs must be pro 
duced anew each time, whereas the written signs 
inscribed on the right material may survive hun 
dreds, even thousands of years. Hence it is that 
the written languages are, upon the whole, much 
better developed than the spoken languages. In 
fact, there are isolated instances in which it may 
be said that the ideal has well-nigh been reached. 

As we have already pointed out, such a case is 
furnished by the written signs of numbers. By a 
systematic manipulation of the ten signs 012345 
6 7 8 9 it is not only possible to co-ordinate a writ 
ten sign with any number whatsoever, but this co 
ordination is strictly unambiguous, that is, each 



The Written Language 91 

number can be written in only one way, and each 
numerical sign has only one numerical significance. 
This has been obtained in the following manner: 

First, a special sign is co-ordinated to each of 
the group of numbers from zero to nine. The 
same signs are co-ordinated with the next group, 
ten to nineteen, containing as many numbers as the 
first. To distinguish the second from the first 
group, the sign one is used as a prefix. The third 
group is marked by the prefixed sign two, and so 
on, until we reach group nine. The following 
group, in accordance with the principle adopted, has 
as its prefix the sign ten, which contains two digits. 
All the succeeding numbers are indicated accord 
ingly. From this the following result is assured : 
First, no number in its sequence escapes designation ; 
second, never is an aggregate sign used for two or 
more different numbers. Both these circumstances 
suffice to secure unambiguity of co-ordination. 

It is known that the system of rotation just de 
scribed is by no means the only possible one. But 
of all systems hitherto tried it is the simplest and 
most logical, so that it has never had a serious rival, 
and the clumsy notation with which the Greeks and 
Romans had to plague themselves in their day was 
immediately crowded out, never to return again 
upon the introduction of the Indo-Arabic notation, 
which has made its way in the same form among all 
the civilized nations and constitutes a uniform part 
of all their written languages. 



92 Logic and Mathematics 

The comparison of the spoken and the written 
languages offers a very illuminating proof of the 
much greater imperfection of the language of 
words. The number 18654 is expressed in the Eng 
lish language by eighteen thousand six hundred and 
fifty-four, that is, the second figure is named first, 
then the first, the third, the fourth, and the fifth. 
In addition, four different designations are used to 
indicate the place of the figures, -teen, -thousand, 
-hundred, and -ty. A more aimless confusion can 
scarcely be conceived. It would be much clearer 
to name the figures simply in their sequence, as 
one-eight-six-five-four. Besides, this would be un 
ambiguous. If we should desire to indicate the 
place value in advance, we could do so in some con 
ventional way, for example, by stating the number 
of digits in advance. This, however, would be 
superfluous, and ordinarily should be omitted.* 

32. Pasigraphy and Sound Writing. There are 
two possibilities for co-ordination between concepts 
and written signs. Either the co-ordination is di 
rect, so that it is only a matter of providing every 
concept with a corresponding sign, or it is indirect, 

*The usual designation of the larger groups, ten, hundred, 
thousand, million, billion, etc., is also quite irrational. If it 
is our object to secure expressions for place values in as few 
words as possible, we find that the numbers of the form 
lo 211 , in which n is a whole number, must receive their own 
names, that is, 10, 100. 10,000, 100,000,000 etc. In this way the 
problem of designating as many numbers as possible by as 
few words as possible is solved. 



Pasigraphy and Sound Writing 93 

the signs serving only the purpose of expressing the 
language sound. In the latter case the written lan 
guage is based entirely upon the sound language, 
and the only problem, comparatively easy to solve, 
is to construct an unambiguous co-ordination be 
tween sound and sign. The Chinese script follows 
the direct process, but all the scripts of the European- 
American civilized peoples are based on the indirect 
process. 

This, it is true, is the case only in ordinary, non- 
scientific language, while for science the European 
nations also have to a large extent built up a direct 
concept writing. One example of this we have 
seen in the number signs. Musical notation fur 
nishes another instance, though by far not so per 
fect. The use of the different keys destroys the 
unambiguous connection between the pitch and the 
note sign, and the signatures placed at the beginning 
of a whole staff have the defect of removing the 
sign from the place where it is applied. Despite 
this imperfection musical notation is quite interna 
tional, and every one who understands European 
music also understands its signs.* 

Fundamentally we need not hesitate to recog 
nize in concept writing or pasigraphy a more com 
plete solution of the problem of sign arrangement. 
Even the very incomplete Chinese pasigraphy ren- 

* It is not difficult to perfect musical notation with a view 
to unambiguity, a thing which would greatly facilitate the 
study of music. 



94 Logic and Mathematics 

clers possible written intercourse, especially for mer 
cantile purposes, between the various East-Asiatic 
peoples who speak some dozens of different lan 
guages. But each language community translates 
the common signs into its own words, just as we do 
in the case of the number signs. But in order that 
such a system of representation should be com 
plete it must fulfil a whole series of conditions for 
which scarcely a remote possibility is to be dis 
cerned at present. 

At first the concepts could simply be taken as 
found in the words and grammatical forms of the 
various languages, and each one provided with an 
arbitrary sign. Such approximately is the Chinese 
system. But a system of that sort entails an ex 
treme burdening of the memory, which results both 
from the great number of words and from the neces 
sity of keeping the signs within certain bounds of 
simplicity. If we consider that the complex con 
cepts are formed according to laws, to a large ex 
tent still unknown, from a relatively small number 
of elementary concepts, we may attempt to build 
up the signs of the complex concepts by the com 
bination of those of the elementary concepts ac 
cording to corresponding rules. Then it would 
only be necessary to learn the signs for the ele 
mentary concepts and the rules of combination in 
order for us to be able to represent all the possible 
concepts. This would provide even for the natural 
enlargement of the concept world, since every new 



Pasigraphy and Sound Writing 95 

elementary concept would receive its sign and would 
then serve as the basis from which to deduce all 
the complex concepts dependent upon it. In fact, 
even should a concept hitherto regarded as ele 
mentary prove to be complex, it would not be dif 
ficult to declare that its sign, like the name of an 
extinct race, is dead, and after the lapse of suf 
ficient time to use it for other purposes. 

The numerical signs offer an excellent example 
for the elucidation of this subject, and at the same 
time serve as a proof that in limited provinces the 
ideal has already been attained. Another very in 
structive example is furnished by the chemical 
formulas, which, though they use the letters of the 
European languages, do not associate with them 
sound concepts, but chemical concepts. Since the 
chemical concepts are co-ordinated with certain let 
ters, it is possible, in the first place, to denote the 
composition of all combinations qualitatively by the 
combination of the corresponding letters. But since 
quantitative composition proceeds according to 
definite relations which are determined by a variety 
of specific numbers peculiar to each element and 
called its combining weight, we need only add to 
the sign of the element the concept of the combin 
ing weight in order to represent in the second place 
the quantitative composition. Further, the mul 
tiples mentioned can also be given. Since, more 
over, there are various substances which, despite 
equal composition, possess different properties, the 



96 Logic and Mathematics 

attempt has been made to express this new mani- 
foldness by the position of the element signs on 
the paper, and in more recent times also by space 
representation. And here, too, rules have been 
worked out in which the scheme affords a close ap 
proach to experience. This example shows how, by 
the constant increase of the complexity of a con 
cept (here the chemical composition), ever greater 
and more manifold demands are made upon the co 
ordinated scheme. The form of expression first 
chosen is not always adequate to keep pace with the 
progress of science. In this case it must be radically 
changed and formed anew to meet the new demands. 
33. Sound Writing. In point of unambiguity of 
co-ordination phonetic writing is far more imper 
fect than concept writing. It is obvious that in 
phonetic writing all the faults already present in 
the co-ordination between concept and sound are 
transferred to the written language. To these are 
added the defects as regards unambiguity occurring 
in co-ordination between sound and sign from which 
no language is free. In some languages, in fact, 
notably in English, these defects amount to a cry 
ing calamity. The principle of unambiguity would 
require that there should never be a doubt as to the 
way in which a spoken word is written, and as little 
doubt as to the way in which a written word is 
spoken. It needs no proof to show how often the 
principle is violated in every language. In the Ger 
man language the same sound is represented by i, v, 



The Science of Language 97 

and ph; in the English by f and ph. And in both 
German and English quite different sounds are as 
sociated with c, g, s, and other letters. The fact 
that orthographic mistakes can be made in the writ 
ing of any language is direct proof of its imperfec 
tion, and the oftener this possibility occurs the more 
imperfect is the language in this respect. We know 
that the spelling reforms begun in Germany more 
than ten years ago and recently in America and 
England, have for their object unambiguity in the 
co-ordination between sign and sound. Still it 
must be admitted that this tendency has not always 
been pursued undeviatingly. A few innovations, in 
fact, undoubtedly represent a step backward. 

34. The Science of Language. A comparison of 
our investigations which we cannot present in de 
tail but only indicate with the science of language 
or philology as taught in the universities and in a 
great number of books, reveals a great difference 
between them. This academic philology makes a 
most exhaustive study of relations, which from the 
point of view of the purpose of language are of no 
consequence whatever, such as most of the rules and 
usages of grammar. A study of this sort must natu 
rally confine itself to a mere determination of 
whether certain individuals or groups of individuals 
have or have not conformed to these rules. Even 
the chief subject of modern comparative philology, 
the study of the relations of the word forms to one 
another and their changes in the course of history, 



98 Logic and Mathematics 

both within the language communities and when 
transferred to other localities, appear to be quite 
useless from the point of view of the theory of co 
ordination. For it is indeed of little moment to us 
to learn by what process of change, as a rule utterly 
superficial, a certain word has come to be co 
ordinated with a concept entirely different from the 
one with which it had been previously co-ordinated. 
Of incomparably greater importance would be in 
vestigations concerning the gradual change of the 
concepts themselves, although by no means as im 
portant as the real study of concepts. To be sure, 
such investigations are much more difficult than the 
study of word forms set down in writing. 

Nevertheless, on account of a historical process, 
which it would lead us too far afield to discuss, an 
idea of such word investigations has been formed 
which is wholly disproportionate to their im 
portance. And if we ask ourselves what part such 
labors have taken in the progress of human civiliza 
tion, we are at a loss for an answer. Students of 
the science of language make a sharp distinction 
between it and the knowledge of language, which 
is regarded as incomparably lower. But while a 
knowledge of language is important in at least one 
respect, in that it presents to us the cultural ma 
terial set down in other languages, or makes them 
accessible in translation to those who do not know 
foreign languages, philology is of no service in 
this respect at all, and the pursuit of it will seem 



The Science of Language 99 

as inconceivably futile to future science as the 
scholasticism of the middle ages seems to us now. 

The unwarranted importance attached to the his 
torical study of language forms is paralleled by the 
equally unwarranted importance ascribed to gram 
matical and orthographic correctness in the use of 
language. This perverse pedantry has been car 
ried to such lengths that it is considered almost dis 
honorable for any one to violate the usual forms of 
his mother tongue, or even of a foreign language, 
like the French. We forget that neither Shake 
speare nor Luther nor Goethe spoke or wrote a 
" correct " English or German, and we forget that it 
cannot be the object of a true cultivation of lan 
guage to preserve as accurately as possible existing 
linguistic usage, with its imperfections, amounting 
at times to absurdities. Its real object lies rather in 
the appropriate development and improvement of 
the language. We have already mentioned the fact 
that in one department, orthography, the true con 
ception of the nature of language and of its develop 
ment is gradually beginning to assert itself. Among 
most nations efforts are being made to improve 
orthography with a view to unambiguity, and when 
once sufficient clearness is had as to the object aimed 
for in spelling, there will be no special difficulty in 
finding the required means to attain it. 

But in all the other departments of language we 
are still almost wholly without a conception of the 
genuine needs. Though the example of the Eng- 



ioo Logic and Mathematics 

lish language proves that we can entirely dispense 
with the manifold co-ordinations in the same 
sentence as appearing in the special plural forms of 
the adjective, verb, pronoun, etc., yet the idea of 
consciously applying to other languages the natural 
process of improvement unconsciously evolved in the 
English language seems not to have occurred even 
to the boldest language reformers. So strongly are 
we all under the domination of the " schoolmaster " 
ideal, that is to say, the ideal of preserving every 
linguistic absurdity and impracticability simply be 
cause it is " good usage." 

A twofold advantage will have been attained by 
the introduction of a universal auxiliary language (p. 
183). Recently the efforts in that direction have 
made considerable progress. In the first place it will 
provide a general means of communication in all mat 
ters of common human interest, especially the sci 
ences. This will mean a saving of energy scarcely to 
be estimated. In the second place, the superstitious 
awe of language and our treatment of it will give 
way to a more approprate evaluation of its technical 
aim. And when by the help of the artificial auxili 
ary language, we shall be able to convince ourselves 
daily how much simpler and completer such a lan 
guage can be made than are the " natural " lan 
guages, then the need will irresistibly assert itself 
to have these languages also participate in its ad 
vantages. The consequences of such progress to 
human intellectual work in general would be ex- 



Continuity 101 

traordinarily great. For it may be asserted that 
philosophy, the most general of all the sciences, has 
hitherto rnade such extremely limited progress only 
because it was compelled to make use of the medium 
of general language. This is made obvious by the 
fact that the science most closely related to it, 
mathematics, has made the greatest progress of all, 
but that this progress began only after it had pro 
cured both in the Indo-Arabic numerals and in the 
algebraic signs a language which actually realizes 
very approximately the ideal of unambiguous co 
ordination between concept and sign. 

35. Continuity. Up to this point our discussions 
have been based on the general concept of the 
thing f that is, of the individual experience differ 
entiated from other experiences. Here the fact of 
being different, which, as a general experience, led 
to the corresponding elementary concept, appeared 
in the foreground in accordance with its generality. 
But in addition to it there is another general fact of 
experience, which has led to just as general a con 
cept. It is the concept of continuity. 

When, for example, we watch the diminution of 
light in our room as it grows dark in the evening, 
we can by no means say that we find it darker at 
the present moment than a moment before. We re 
quire a perceptibly long time to be able to say with 
certainty that it is now darker than before, and 
throughout the whole time we have never felt the 
increase of darkness from moment to moment, al- 



IO2 Logic and Mathematics 

though theoretically we are absolutely convinced 
that this is the correct conception of the process. 

This peculiar experience, our failure to perceive 
individual parts of a change, the reality of which 
we realize when the difference reaches a certain de 
gree, is very general, and, like memory, is based 
upon a fundamental physiological fact. It has al 
ready been noted by Herbart, but its significance 
was first recognized by Fechner, and has since then 
become generally known in physiology and psychol 
ogy under the name of threshold. Next to memory 
the threshold determines the fundamental lines of 
our psychic life. 

The threshold therefore means that whatever 
state we are in a certain Unite amount of difference 
or change must be stepped over before we can per 
ceive the difference or change. This peculiarity ap 
pears in all our states or experiences. We have al 
ready given an example for the phenomena of light 
and darkness. The same is true of differences in 
color and of our judgments as to tone pitch and 
tone strength. Even the transition from feeling- 
well to feeling ill is usually imperceptible, and it is 
only when the change occurs in a very brief time 
that we become conscious of it. 

C *""*"* - LW^^ >>> 

The physical causes of these psychic phenomena 
need be indicated only in brief. In all our experi 
ences an existing chemico-physical state in our 
sense organs and in the central organ undergoes a 
change. Now experiments with physical apparatus 



Continuity 103 

have shown that such a process always requires a 
finite, though sometimes a very small, quantity of 
work, or, generally speaking, energy, before it can 
be brought about at all. Even the finest scale, sensi 
tive to a millionth of a gram, remains stationary 
when only a tenth of a millionth is placed upon it, 
although we can see a body of such minute weight 
under the microscope. In the same way it requires 
a definite expenditure of energy in order to bring 
the sense organs, or the central organ, into action, 
and all stimuli less than this limit or threshold pro 
duce no experience of their presence. 

By this the difficult concept of continuity is 
evoked in our experience. The transition from the 
light of day to the darkness of evening proceeds con 
tinuously, that is, at no point of the whole transition 
do we notice that the state just passed is different 
from the present one, while the difference over a 
wider extent of the experience is unmistakable. If 
we wish to bring vividly to our minds the contradic 
tion to other habits of thought which this involves, 
we need only to represent to ourselves the following 
instance. I will compare the thing A at a certain 
time with the thing B, which is so constructed that 
though objectively different from A, the difference 
has not yet reached the threshold. From experi 
ence, therefore, I must take A to be equal to B. 
Then I compare B with a thing C, which is ob 
jectively different from B in the same way as A is 
from B, though here, too, the difference is still 



104 Logic and Mathematics 

within the threshold, though very near it. I shall 
also have to take B as equal to C. But now if I 
compare A directly with C, the sum of the two dif 
ferences oversteps the threshold value, and I find 
that A is different from C. This, then, is a con 
tradiction of the fundamental principle that if A=B 
and B=C, A=C. This principle is valid for counted 
things, which, in consequence, are discontinuous, but 
not for continuous things susceptible by our senses. 
If in spite of this it is applied to continuous things 
or magnitudes in the narrower sense, we must bear 
in mind that it is just as much a case of an extrapola,- 
tion to the non-existing ideal instance (p. 46) as 
in the case of the other general principles, which, 
though they are derived from experience, neverthe 
less, for practical purposes, transcend experience in 
their use. 

The examples cited above prove also that these 
relations are by no means confined to the judgments 
we derive on the basis of immediate sensations. 
When by means of the scale we compare three 
weights, the differences of which lie within the limit 
of its sensitiveness but approach closely to it, w r e 
can arrive in a purely empirical and objective way 
also at the contradiction A=B, B=C, but A^C. In 
weight and measurement, therefore, we hold fast to 
the principle that the relations cited have no claim 
to validity outside the limit of their possible er 
rors. Accordingly, though the non-equation of 
can be observed, the difference of both values 



Continuity 105 

cannot be greater than at utmost the sum of the two 
threshold values. 

These considerations also give us a means of ap 
praising the oft-repeated statement that in contra 
distinction to the physical laws the mathematical 
laws are absolutely accurate. The mathematical 
laws do not refer to real things, but to imaginary 
ideal limit cases. Consequently they cannot be 
tested by experience at all, and the demands science 
makes on them lie in quite a different sphere. Their 
nature must be such that experience should approx 
imate them infinitely, if certain definite well-known 
postulates are to be more and more fulfilled, and 
that the various abstractions and idealizations 
should be so chosen as not to contradict one an 
other. Such contradictions have by no means al 
ways been avoided. But we must not regard them 
as inherent in the inner organization of our mind, 
as Kant did. These contradictions spring from 
careless handling of the concept technique, by which 
postulates elsewhere rejected are treated as valid. 
We have already come across an instance of such 
relations in the application of the concept of equal 
ity to unlimited groups (p. 84). 

We must be guided by the same rules of precau 
tion in answering the question whether the things felt 
as continuous for example, space and time are 
" truly " continuous, or whether in the last analysis 
they must not be conceived of as discontinuous. The 
various sense organs, and still more, the various 



io6 Logic and Mathematics 

physical apparatus with which we examine given 
states, are of very varying degrees of " sensibility/* 
that is, the threshold for distinguishing the differ 
ences may be of very different magnitudes. There 
fore, a thing which is discontinuous for a sensitive 
apparatus will behave as if it were continuous with a 
less sensitive apparatus. Accordingly, we shall find 
so many the more things continuous the less sharply 
developed our ability is to differentiate. 

While this circumstance makes it possible that 
we should regard discontinuous things as continu 
ous, time relations in certain circumstances produce 
the opposite effect. Even if in a process the change 
is continuous but very rapid, and the new state re 
mains unchanged for a certain time, we easily con 
ceive of this sequence as discontinuous. We cannot 
resist this view of the process when the change oc 
curs in a shorter time than the threshold time of 
our mind for each step in the process. But since 
this threshold changes with our general condition, 
one and the same process can appear to us both con 
tinuous and discontinuous according to circum 
stances. Here, therefore, we have a cause through 
the operation of which, with advancing knowledge, 
more and more things will become recognized as 
continuous. 

Now if we turn to experience, we find, as the 
sum total of our knowledge, that for the sake of 
expediency we approach everything with the pre 
sumption that it is continuous. This aggregate ex- 



Measurement 107 

perience finds its expression in such sayings as " Na 
ture makes no jumps," and similar proverbial gen 
eralizations. But we must emphasize the fact once 
more that in deciding matters in this way we deal 
solely with questions of expediency, not with ques 
tions of the nature of our mental capacity. 

36. Measurement. Measuring is in a certain way 
the opposite of counting. While, in counting, the 
things are regarded in advance as individual, and 
the group, therefore, is a body compounded of dis 
continuous elements, measuring, on the other hand, 
consists in co-ordinating numbers with continuous 
things, that is, in applying to continuous things a 
concept formed upon the hypothesis of discontinuity. 

It lies in the nature of such a problem that the dif 
ficulty of adaptation must crop out somewhere in 
the course of its attempted solution. This is actu 
ally shown by the fact that measurement proves to 
be an unconcluded and inconcludable operation. If, 
in spite of this, measurement may and must justly 
be denoted as one of the most important advances 
in human thought, it follows that those fundamental 
difficulties can practically be rendered harmless. 

Let us picture to ourselves some process of meas 
urement for example, the determination of the 
length of a strip of paper. We place a rule divided 
into millimeters (or some other unit) on the strip, 
and then we determine the unit-mark at which the 
strip ends. It turns out that the strip does not end 
exactly at a unit-mark, but between two unit-marks. 



io8 Logic and Mathematics 

And even if the rule is provided with divisions ten 
or a hundred times finer, the case remains the same. 
In most cases a microscopic examination will show 
that the end of the strip does not coincide with a 
division. All that can be said, therefore, is that 
the length must lie between n and n + i units, and 
even if a definite number is given, the scientifically 
trained person will supplement this number by the 
sign f, in which / denotes the possible errors, that 
is, the limit within which the given number may be 
false. 

We see at once how the characteristic concept of 
threshold, which has led to the conception of the 
continuous, immediately asserts itself when in con 
nection with discontinuous numbers. The adapta 
tion of the threshold to numbers can be carried as 
far as it is possible to reduce the threshold, but the 
latter can never be made to disappear entirely. 

The significance of measurement therefore lies 
in the fact that it applies the operation of counting 
with all its advantages (see p. 85) to continuous 
things, which as such do not at first lend themselves 
to enumeration. By the application of the unit meas- 
sure a discontinuity is at first artificially established 
through dividing the thing into pieces, each piece 
equal to the unit, or imagining it to be so divided. 
Then we count the pieces. When a quantity of 
liquid is measured with a liter this general process 
is carried out physically. In all other less direct 
methods of measurement the physical process is sub- 



The Function 109 

stituted by an easier process equally good. Thus, 
in the example of the strip of paper we need not 
cut it up into pieces a millimeter in length. The 
divided rule is available for comparing the length 
of any number of millimeters that happen to come 
under consideration, and we need only read off from 
the figures on the rule the quantity of millimeters 
equal to the length of the strip, in order to infer that 
the strip can be cut up into an equal number of 
pieces each a millimeter in length. 

After it has been made possible to count continu 
ous things in this way, the numeration of them can 
then be subjected to all the mathematical operations 
first developed only for discrete, directly countable 
things. When we reflect that our knowledge of 
things has given them to us preponderatingly as 
continuous, we at once see what an important step 
forward has been made through the invention of 
measurement in the intellectual domination of our 
experience. 

37. The Function. The concept of continuity 
makes possible the development of another concept 
of greater universality, which can be characterized 
as an extension of the concept of causation (p. 31). 
The latter is an expression of the experience, if A 
is, B is also, in which A is understood to be a 
definite thing at first conceived of as immutable. 
Now it may happen that A is not immutable, but 
represents a concept with continuously changing 
characteristics. Then, as a rule, B will also be of 



no Logic and Mathematics 

that nature, so that every special value or state of B 
corresponds to every special value or state of A. 

Here, in place of the reciprocal relation of two 
definite things, we have the reciprocal relation of 
two more or less extended groups of similar things. 
If these things are continuous, as is assumed here 
(and which is extremely often the case), both groups 
or series, even though they are finite, contain an 
endless quantity of individual cases. Such a rela 
tion between two variable things is called a func 
tion. Although this concept is used chiefly for the 
reciprocal relation of continuous things, there is 
nothing to hinder its application to discrete things, 
and accordingly we distinguish between continuous 
and discontinuous functions. 

The intellectual progress involved in the concep 
tion of the reciprocal relation of entire series or 
groups to one another, as distinguished from the 
conception of the relations between individual 
things, is of the utmost importance and in the most 
expressive manner characterizes the difference be 
tween modern scientific thought and ancient 
thought. Ancient geometry, for example, knew 
only the cases of the acute, right, and obtuse angled 
triangle, and treated them separately, while the mod 
ern geometrician represents the side of the triangle 
as starting from the angle zero and traversing the 
entire field of possible angles. Accordingly, unlike 
his colleague of old, he does not ask for the par 
ticular principles bearing upon these particular 



The Function in 

cases, but he asks in what continuous relation do 
the sides and angles stand to one another, and he 
lets the particular cases develop from out of one 
another. In this way he attains a much profounder 
and more effectual insight into the whole of the ex 
isting relations. 

It is in mathematics in especial that the introduc 
tion of the concept of continuity and of the func 
tion concept arising from it has exercised an ex 
traordinarily deep influence. The so-called Higher 
Analysis, or Infinitesimal Analysis, was the first 
result of this radical advance, and the Theory of 
Functions, in the most general sense, was the later re 
sult. This progress rests on the fact that the mag 
nitudes appearing in the mathematical formulas 
were no longer regarded as certain definite values 
(or values to be arbitrarily determined), but as 
variable, that is, values which may range through 
all possible quantities. If we represent the rela 
tion between two things by the formula B=f(A), 
expressed in spoken language by B is a function of 
A, then in the old conception A and B are each in 
dividual things, while in the modern conception A 
and B represent an inexhaustible series of possibil 
ities embracing every conceivable individual case 
that may be co-ordinated with a corresponding 
case. 

Herein lies the essential advantage of the con 
cept of continuity. It is true that it also introduces 
into calculation the above-mentioned contradictions 



H2 Logic and Mathematics 

which crop up in the ever-recurring discussions con 
cerning the infinitely great and the infinitely small. 
The system introduced by Leibnitz of calculating 
with differentials, that is, with infinitely small quan 
tities, which in most relations, however, still pre 
serve the character of finite quantities from which 
they are considered to have been derived, has proved 
to be as fruitful of practical results as it is difficult 
of intellectual mastery. We can best conceive of 
these differentials as the expression of the law of 
the threshold, which law gave rise to, or made pos 
sible, the relation between the continuous and the 
discrete. 

38. The Application of the Functional Relation. 
I have already shown (p. 34) how the first formula 
tion of a causal relation which experience yields 
can be purified and elaborated by the multiplica 
tion of the experience. The method described was 
based upon the fact that the necessary and ade 
quate factors of the result were obtained by elim 
inating successively from the " cause " the various 
factors of which its concept was or could be com 
pounded, and by concluding from the result, that is, 
the presence or absence of the " effect," as to the 
necessity or superfluity of each factor. 

Obviously the application of this process pre 
supposes the possibility of eliminating each factor 
in turn. Very often it is not possible, and then 
in place of the inadequate method of the individual 
case the method of the continuous functional rela- 



The Law of Continuity 113 

tion steps in with its infinitely greater effectiveness. 
If in most cases we cannot eliminate the factors one 
by one, there are very few instances in which it is 
not possible to change them, or to observe the result 
in the automatically changed values of the factors. 
But then we have the principle that for the causal re 
lation all such factors are essential the cfiange of 
which involves a change of the result. 

It is clear that this signifies a generalization of 
the former and more limited method. For the elim 
ination of the factor means that its value is reduced 
to zero. But now it is no longer necessary to go to 
this extreme limit; it suffices merely to influence in 
some way the factor to be investigated. 

It is true that here the difference in the result 
cannot be expressed with a " yes " or a " no," as 
before. It can only be said that it has changed 
partly, more or less. From this it can be seen that 
the application of this process requires more re 
fined methods of observation, especially for measur 
ing, that is, for determining values or magnitudes. 
On the other hand, we must recognize how much 
deeper we can penetrate into the knowledge of 
things by the application of the measuring process. 
Each advance in precision of measurement signifies 
the discovery of a new stratum of scientific truth 
previously inaccessible. 

39. The Law of Continuity. From the fact that 
natural phenomena in general proceed continuously 
we can deduce a number of important and generally 



H4 Logic and Mathematics 

applicable conclusions which are constantly used 
for the development of science. 

When a relation of two continuously varying 
values of the form A=f (B) is conjectured, we con 
vince ourselves of its truth by observing for dif 
ferent values of A the corresponding values of B, 
or reversely. If we find that changes in the one 
correspond to changes in the other, the existence of 
such a relation is proved, at first only for the ob 
served values, though we never hesitate to conclude 
that for the values of A lying between the observed 
values, but themselves not yet observed, the corre 
sponding values of B will also lie between the ob 
served values. For example, if the temperature 
at a given place has been observed at intervals of 
two hours, we assume without hesitancy that 
in the hours between when no observations were 
made, the values lie between the observed values. 
If we indicate the time in the usual manner by hori 
zontal lines and the temperature for the general 
periods of time by longitudinal lines, the law of 
continuity asserts that all these temperature points 
lie in a steady line, so that when a number of 
points lying sufficiently near one another is known, 
the points between can be derived from the steady 
line which may be drawn through the known points. 
This very commonly applied process will yield the 
more accurate results the nearer the known points 
are to one another, and the simpler the line. 

The application of the law of continuity or steadi- 



The Law of Continuity 115 

ness, therefore, means no less than that it is possible, 
from a finite, frequently not even a very large, num 
ber of individual results, to obtain the means of 
predicting the result for an infinitely large number 
of unexamined cases. The instrument derived 
from this law, therefore, is an eminently scientific 
one. 

The value of this instrument is still greater if it 
succeeds in expressing the relation A= = f(B) in 
strict mathematical form. First, the result of the 
determination of a number of individual values of 
that function is represented as a table of co 
ordinated values. By the graphic process above de 
scribed, or by its equivalent, the mathematical 
process of interpolation, this table is so extended 
that it also supplies all the intermediate values. 
But this is still a case of a mechanical co-ordination 
of the corresponding values. Often we succeed, 
especially in the relation of simple or pure concepts, 
in finding a general mathematical rule by which the 
magnitude A can be derived from the magnitude B, 
and reversely. This is the only instance in which 
we speak of a natural law in the quantitative 
sense. 

Thus, for example, we can observe what volume 
a given quantity of air occupies when successively 
subjected to different pressures. If we arrange all 
these values together in a table, we can also cal 
culate the volume for all the intermediate 
pressures. But on close inspection of the cor- 



ii6 Logic and Mathematics 

responding numbers of pressure and volume we 
notice that they are in inverse ratio, or that when 
multiplied by one another their products will be the 
same. If we denote the space by v and the pressure 
by p, this fact assumes the mathematical form 
p. v=K, in which K is a definite number depending 
upon the quantity of air, the unit of pressure, etc., 
but remaining unchanged in an experimental series 
in which these factors stay the same. The general 
functional equation A=f (B) becomes the definite 

Tjr 

p= . And this formula enables us to determine 

by a simple calculation the volume for any degree 
of pressure, provided the value of K has been once 
ascertained by experiment. 

At first we have a right to such a calculation only 
within the province in which the experiments have 
been made, and the simple mathematical expression 
of the natural law has for the time being no 
further significance than that of a specially con 
venient rule for interpolation. But such a form im 
mediately evokes a question which demands an ex 
perimental answer. How far can the form be ex 
tended ? That there must be a limit is to be directly 
inferred from the consideration of the formula it 
self. For if we let p=o, then v^infinity, both 
of which lie beyond the field of possible experi 
ence. 

Similar considerations obtain in all such mathe 
matically formulated natural laws, and each time, 



The Law of Continuity 117 

therefore, we must ask what the range of validity 
of such an expression is, and answer the question 
by experiment. 

While in this discussion the mathematically for 
mulated natural law seems to have the nature only 
of a convenient formula of interpolation, we are 
nevertheless in the habit of regarding the discovery 
of such a formula as a great intellectual accom 
plishment, which so impresses us that we frequently 
call it by the name of the discoverer. Now, 
wherein lies the more significant value of such 
formulations ? 

It lies in the fact that simple formulas are dis 
covered only when the conceptual analysis of the 
phenomenon has advanced far enough. The very 
simplicity of the formula shows that the concept 
formation which is at the basis of it is especially 
serviceable. In Ptolemy s theory of the motion of 
the planets the means for calculating their positions 
in advance was given just as in the theory of Coper 
nicus. But Ptolemy s theory was based on the as 
sumption that the earth stands still, and that the sun 
and the other planets move. The assumption that 
the sun stands still and that the earth and the other 
planets move greatly facilitates the calculation of 
the position of the planets. In this Jay the primary 
value of the advance made by Copernicus. It was 
not until much later that it was found that a num 
ber of other actual relations could be represented 
much more fittingly by means of the same hypoth- 



ii8 Logic and Mathematics 

esis, and thus the Copernican theory has come to 
be generally recognized and applied. 

The significance of the law of continuity and its 
field of application have by no means been ex 
hausted by what has been said above. But later 
we shall have a number of occasions to point out its 
application in special instances, and so cause its 
use to become a steady mental habit with the be 
ginner in scientific research. 

40. Time and Space. Time and space are two 
very general concepts, though without doubt not 
elementary concepts. For besides the elementary 
concept of continuity which both contain, time has 
the further character of being one-seried or one- 
dimensional, of not admitting of the possibility of 
return -to a past point of time (absence of double 
points) and of absolute onesidedness, that is, of the 
fundamental difference between before and after. 
This last quality is the very one not found in the 
space concept, which is in every sense symmetrical. 
On the other hand, owing to the three dimensions 
it has a threefold manifoldness. 

That despite this radical distinction in the prop 
erties of space and time all of our experiences can 
be expressed or represented within the concepts of 
space and time^ is very clear proof that experience 
is much more limited than the formal manifoldness 
of the conceivable. In this sense space and time 
can be conceived as natural laws which may be ap 
plied to all our experiences. Here at the same 



Time and Space 119 

time the subjective-human element of the natural law 
becomes very clear. 

The properties of time are of so simple and ob 
vious a nature that there is no special science of 
time. What we need to know about it appears as 
part of physics, especially of mechanics. Never 
theless time plays an essential role in phoronomy, a 
subject which we shall consider presently. In pho 
ronomy, however, time appears only in its simplest 
form as a one-seried continuous manifoldness. 

As for space, the presence of the three dimensions 
conditions a great manifoldness of possible rela 
tions, and hence the existence of a very extensive 
science of bodies in space, of geometry. Geometry 
is divided into various parts depending upon whether 
or not the concept of measurement enters. When 
dealing with purely spacial relations apart from the 
concept of measurement it is called geometry of 
position. In order to introduce the element of 
measurement a certain hypothesis is necessary 
which is undemonstrable, and therefore appears to 
be arbitrary and can be justified only because it is 
the simplest of all possible hypotheses. This 
hypothesis takes for granted that a rigid body 
can be moved in all directions in space without 
changing in measure. Or, to state the inverse of 
this hypothesis, in space those parts are called equal 
which a rigid body occupies, no matter how it is 
moved about. 

We are not conscious of the extreme arbitrariness 



120 Logic and Mathematics 

of this assumption simply because we have become 
accustomed to it in school. But if we reflect that in 
daily experience the space occupied by a rigid body, 
say a stick, seems to the eye to undergo radical 
changes as it shifts its position in space and that we 
can maintain that hypothesis only by declaring these 
changes to be " apparent," we recognize the ar 
bitrariness which really resides in that assumption. 
We could represent all the relations just as well if 
we were to assume that those changes are real, and 
that they are successively undone when we restore 
the stick to its former relation to our eye. But 
though such a conception is fundamentally practi 
cable in so far as it deals merely with the space pic 
ture of the stick, we nevertheless find that it would 
lead to such extreme complications with regard to 
other relations ( for example, the fact that the weight 
of the stick is not affected by the change of the optic 
picture) that we do better if we adhere to the usual 
assumption that the optical changes are merely ap 
parent. 

In this connection we learn what an enormous in 
fluence the various parts of experience exert upon 
one another in the development of science. In every 
special generalization of experiences, that is, in every 
individual scientific theory, our aim is not only to 
generalize this special group of experiences in them 
selves, but at the same time to join such other ex 
periences to them as expedience demands. If the 
effect of this necessity is on the one hand to render 



Time and Space 121 

the elaboration of an appropriate theory more dif 
ficult, it has on the other hand the great advantage 
of affording a choice among several theories of 
apparently like value, and thus making possible a 
more precise notion of the reality. For example, 
for the understanding of the mutual movements of 
the sun and the earth it is the same whether we as 
sume that the sun moves about the earth or the 
earth about the sun. It is not until we try to rep 
resent theoretically the position of the other planets 
that we see the economic advantage of the second 
conception, and facts like Foucault s experiment 
with a pendulum can be represented only according 
to this second conception in our present state of 
knowledge. 

Likewise, the assumption on which scientific 
geometry goes, that space has the same properties 
in all directions, conflicts with immediate experi 
ence. In immediate experience we make a sharp 
distinction between below and above, although we 
are prepared to admit the " homogeneity " of space 
in the horizontal direction. This is due, as physics 
teaches, to the fact that we are placed in a field of 
gravitation which acts only from above downward 
and which permits free horizontal turnings, although 
it imparts a characteristic difference to the third di 
rection. Since considerations of another kind en 
able us to place ourselves in a position in which 
we ignore this field of gravitation in the investiga 
tion of space, geometry abstracts this element and 



122 Logic and Mathematics 

disregards the corresponding manifoldness. In the 
theory of the gravitation potential, on the other 
hand, this very manifoldness is made the subject of 
scientific investigation. 

The common application of the concepts of space 
and time results in the concept of motion, the science 
of which is called phoronomics. In order to make 
this new variable subject to measurement we must 
arrive at an agreement or convention as to the way 
in which to measure time. For since past time can 
never be reproduced we actually experience only 
unextended moments, and have no means of recog 
nizing or defining the equality of two periods of 
time by placing them side by side, as we can in 
the case of spacial magnitudes. We help ourselves 
by saying that in uninfluenced motions equal periods 
of time must correspond to the equal changes in 
space. We regard the rotation of the earth on its 
axis and its revolution about the sun as such un 
influenced motions. The two depend upon dis 
similar conditions, and the empirical fact that the 
relation of the two motions, or the relation between 
the day and the year, remains practically the same, 
sustains that assumption, and at the same time 
shows the expediency of the given definition of 
time. 

Analytic geometry, the application of algebra to 
geometric relations, occupies a noteworthy position, 
from the point of view of method, in the science of 
space. It yields geometric results by means of cal- 



Time and Space 123 

dilation, that is, by the application of the algebraic 
material of symbols we can obtain data concerning 
unknown spacial relations. An explanation is 
necessary of how by a method apparently so ex 
traneous such results as these can be attained. 

The answer lies again in the general principle of 
co-ordination, which in this very case receives a 
particularly cogent illustration. Three algebraic 
signs, x, y, and z, are co-ordinated with the three 
variable dimensions of space. First, the same in 
dependent and constant variability is ascribed to 
these signs, and, further, the same mutual relations 
are assumed to subsist between them as actually 
exist between the three-spacial dimensions. In 
other words, precisely the same kind of manifold- 
ness is imparted to these algebraic signs as the 
spacial dimensions possess to which they are co 
ordinated, and we may therefore expect that all the 
conclusions arising from these assumptions will find 
their corresponding parts in the spacial manifold- 
ness. Accordingly, a co-ordinated spacial relation 
corresponds to every change of those algebraic for 
mulas resulting from calculation, and if such 
changes lead to an algebraically simple form, then 
the spacial form corresponding to it must show 
an analogous simplicity. Here, therefore, we have 
a case such as was described under simpler condi 
tions on p. 86 of operations undertaken with one 
group and repeated correspondingly in the co 
ordinated group. And it is only the great difference 



124 Logic and Mathematics 

in the things of which in this case the two groups 
are composed spacial relations on the one side 
and algebraic signs on the other that creates the 
impression of astonishment which was felt very 
strongly at the invention of this method, and which 
is still felt by students with talent for mathematics 
when they first become acquainted with analytical 
geometry. 

41. Recapitulation. Before we proceed to con 
sider the fundamentals of other sciences, it is well to 
make a general resume of the field so far traversed. 
Since the later sciences, as we have already observed, 
make use of the entire apparatus of the earlier sci 
ences, the mastery of them must be assured in 
order to render their special application possible. 

This does not mean that one must have complete 
command of the entire range of those earlier sciences 
in order to pursue a later one. Mere human lim 
itations would prevent the fulfilment of such a de 
mand. As a matter of fact, successful work can 
be done in one of the later sciences even if only 
the most general features of the earlier ones have 
been clearly grasped. Nevertheless, the rapidity and 
certainty of the results are very considerably in 
creased by a more thorough knowledge of the earlier 
sciences, and the investigator, accordingly, should 
seek a middle road between the danger of insuf 
ficient preparation for his special science and the 
danger of never getting to it from sheer prepara 
tion. In any circumstances he must be prepared al- 



Recapitulation 125 

ways, even though it be in later age, to acquire 
those fundamental aids so soon as he feels the need 
of them for carrying out any special work. It is 
generally acceded that without logic the adequate 
pursuit of science is impossible. Nevertheless, the 
opinion is widely current, even among men of sci 
ence, that everybody has command of the needful 
logic without having studied it. No more than a 
man can learn of himself to use the calculus, even 
if he may have discovered unaided some of its ele 
mentary principles, can he acquire certainty and 
readiness in the use of the logical rules generally 
necessary, unless he has made the necessary studies. 
It is true that the scientific works of the great 
pioneers and leaders in the special sciences furnish 
practical examples of such logical activity. But 
complete freedom and security are acquired only on 
the basis of conscious knowledge. 

We have now seen how, from the physiological 
construction of our mental apparatus, the process 
of concept formation and the experience of concept 
connections are the basis of the whole of mental 
life. The laws of the mutual interaction of the 
most general or elementary concepts operated in the 
formation of the concepts, thing, group, co-ordina 
tion. Here were found the fundamentals of logic 
or the science of concepts. A special process of ab 
straction yielded the concept of number, and with 
it the corresponding field of mathematics, arithmetic, 
algebra, and the theory of numbers. 



126 Logic and Mathematics 

By means of the second fundamental fact of 
physiology, the threshold, another elementary fact 
was explained, that of continuity. The co-ordina 
tion of individual things under the influence of this 
concept was expanded into the co-ordination of con 
tinuous phenomena-series, and yielded the corre 
spondingly more general concept of the function. 
From the application of the number concept to con 
tinuous things, the idea of measurement resulted. 
In mathematics the concept of continuity led to 
higher analysis and the theory of functions. Finally, 
the concept of continuity proved to be an inex 
haustible aid for the extension of scientific knowl 
edge and for the formulation of natural laws in 
mathematical form. 



PART III 
THE PHYSICAL SCIENCES 

42. General. In the formal sciences we began 
the specialization of the object from the most gen 
eral concept of thing conceivable, possessing no 
other characteristic attribute than its capability of 
being distinguished from other things ; and we car 
ried the specialization so far that we could follow 
in its movements an object definite as to time and 
space. This object, to be sure, was defined only in 
that it occupied a definite space, and accordingly had 
a definite form. As a matter of fact, the spacial 
thing of geometry and phoronomy reveals no 
further attributes. 

It is here that the physical sciences enter into 
their dominion one after the other, and fill the empty 
space of the geometric thing with definite at 
tributes. These are the secondary qualities of 
Locke, of which he assumed that they do not be 
long so much to the bodies themselves as that they 
merely appear to us so on account of the nature 
of our human sense organs. Now that our knowl 
edge concerning the nature of those properties as 
well as the structure of our sense organs is much 

137 



ia8 The Physical Sciences 

more thorough, we have more definite ideas also of 
the subjective part of the corresponding experiences, 
and in a large measure are able to separate it from 
the objective part. 

All properties which physical bodies in contra 
distinction to geometric bodies possess can be traced 
back to a fundamental concept, which, in conjunc 
tion with the concepts explained in the former chap 
ter, serves to characterize and distinguish the phys 
ical structure. For example, the fact that we can 
distinguish cubes of equal size but of different ma 
terial, different temperature, and different luminos 
ity, can be traced back always and entirely to the 
different kinds of energy acting in the geometric 
space in question. The concept of energy, there 
fore, plays approximately the same role in the phys 
ical sciences as the concept of thing in the formal 
sciences, and the essentials of this new field of sci 
ence are the comprehensive knowledge and de 
velopment of this concept. Because of its great im 
portance it has long been known and applied in in 
dividual forms. But the systematization of the 
physical sciences relative to energy is a matter of 
only recent date. 

43. Mechanics. Recently many scientists have 
taken exception to the traditional division of 
mechanics into statics, or the science of equilibrium, 
and dynamics, or the science of motion, because it 
does not correspond to the essence of the thing, 
equilibrium being only the limit-case of motion. 



Mechanics 129 

However, the classic presentations of this science 
are based on that division, so that it must express 
an essential difference. This difference we can 
clearly recognize through the application of the con 
cept of energy to mechanics. We then learn that 
statics is the science of work, or the energy of posi 
tion, and that dynamics is the science of living 
force, or of the energy of motion. 

By work in the mechanical sense we mean the 
. expenditure of force required for the locomotion 
of physical bodies. While a cube of lead is geo 
metrically equal to a cube of glass, we experience 
a great difference between them when we lift them 
from the floor to a table. We call the cube of lead 
heavier than the glass cube, and we find it requires 
more work to raise the former than the latter. For 
psychologic reasons this judgment becomes especially 
clear when the work required to lift the lead cube 
marks the limit of our physical capacity. 

Work depends not only upon the difference de 
scribed above, but also upon the distance through 
which it is exerted. It increases in proportion as 
the distance increases. In mechanics work is propor 
tional both to the distance and to that peculiar prop 
erty which in the given example we call weight. 
But a more general concept has been formed for that 
property in the mechanical sense, called force, of 
which weight constitutes but a special instance. 
Whenever there is a resistance combined with a 
change of place we speak of a force, and the 



130 The Physical Sciences 

product of the force and the distance we call 
work. 

The cause of this kind of concept formation is 
the following: There are a great number of differ 
ent machines, all of them possessing the peculiarity 
that work can be put into them at a definite place 
and taken out at another place. Now, centuries of 
experience have shown that it is impossible to ob- 
* tain more work from such mechanical machines than 
has been put into them. As a matter of fact, the 
t work obtained is always less than the work put in, 
and the two approach equality as the machine ap 
proaches perfection. It is to such ideal machines, 
therefore, that the law of the conservation of work 
applies. This law states that, though a given quan 
tity of work may be changed in the most manifold 
ways as to direction, force, etc., it is impossible to 
change its quantity. 

The reason we can judge of this fact with such 
certainty is because for many centuries a number 
of the ablest mechanicians have sought for a solu 
tion of the problem of perpetual motion, that is, for 
the construction of a machine from which more 
work can be gotten than is put into it. All such 
attempts have failed. But the positive result se 
cured from these apparently futile efforts is the law 
of the conservation of work. The greatness and 
importance of this result will become apparent in the 
further course of our study. 

Here for the first time we meet with a law ex- 



Mechanics 131 

pressing the quantitative conservation of a thing, 
which may none the less undergo the most varied 
qualitative changes. With the knowledge of this 
fact we involuntarily combine the notion that it is 
the "same" thing that passes through all these 
transformations, and that it only changes its out 
ward form without being changed in its essence. 
Such ideas, it is true, are widespread, but they have 
a very doubtful side to them, since they correspond 
to no distinct concept. If we want to call the quan 
titative magnitude of the product of the force and 
distance the " essence "of work, and the determina 
tion of the force and the distance according to mag 
nitude and direction, which come under considera 
tion for each special value, as its " form," then, of 
course, there is no objection to be made to mere 
nomenclature. But we must bear in mind that the 
difference obtaining here lies exclusively in the fact 
that the amount of work measured quantitatively re 
mains unchanged, while its factors undergo simul 
taneous and opposite changes. 

This discovery, that there is a magnitude which 
can be quantitatively determined, and which, as ex 
perience shows, remains unchanged, however much 
its factors may change, invariably results not only 
in a very simple and clear formulation of the cor 
responding natural law, but also corresponds to the 
general tendency of the human mind to work out 
conceptually " the permanent in change." If, in ac 
cordance with the word-sense, we denote everything 



132 The Physical Sciences 

which persists under changing conditions by the 
name of substance, we encounter in work the first 

* substance of which we have attained knowledge in 
our scientific journeys. In the history of the evolu 
tion of human thought this substance has been pre- 

t ceded by others, especially by the weight and mass 
of ponderable bodies (which are also subject to a 
law of conservation), so that at present we are in 
clined to connect with the word substance a tacit 
secondary sense of ponderability. But this is a 
remnant of the still very widely spread mechanistic 
theory of the universe, which, though it has almost 
finished its role in physics, will presumably continue 
to persist for a long time to come in the popularly 
scientific consciousness in accordance with the laws 
of collective thought. 

44. Kinetic Energy. The law of the conserva 
tion of work is by no means true of all cases in 
which work is expended or converted, but, as has 
been said, only of ideal machines, that is, of such 
cases which do not exist in reality. But while in 
imperfect machines there is at least an approxima 
tion to this law, there are besides countless normal 
cases in which we cannot even speak of an approx 
imation. When, for example, a stone falls to the 
ground from a certain height, a certain quantity of 
work is expended, which is equal to that by means 
of which the stone can be raised again to its original 
height. This quantity of work apparently disap 
pears entirely when the stone remains lying on the 



Kinetic Energy 133 

ground. We shall discuss this case later. Or the 
falling of the stone can be so guided that it can lift 
itself again. This happens, for instance, when, by 
fastening the stone to a thread, it is forced to 
move in a curved path, or to perform pendular 
oscillations. In that case, it is true, the stone will 
fall to the lowest point which the thread permits, 
and so will there have lost its work without having 
done any other work in the meantime. But it has 
entered a condition by virtue of which it raises itself 
again, so that (as before, only in the ideal limit- 
case) it reaches its former height, and so has lost 
no work. For this moment, too, then, the law of 
the conservation of work obtains. But in the mean 
time new relations have arisen. 

What distinguishes the stone moving like a pen 
dulum from the stone which simply falls is, that at 
its lowest point it has not remained lying still, but 
possesses a certain velocity. By means of this it 
lifts itself again, and after it has reached its former 
height, it has lost its velocity. Therefore, there is 
a reciprocal relation between the work which it loses 
and the velocity which it gains, and the question 
may therefore be put, How can this relation be rep 
resented mathematically? Experience teaches that 
in every such case a function of the velocity and of 
another property of the body, called mass, can be 
established in such a way that this function, called 
the kinetic energy of the body, increases precisely as 
much as the amount of work the body has ex- 



134 The Physical Sciences 

pended, and vice versa. The sum of the kinetic 
, energy of the body and of the work is therefore 
constant, and the clearest mode of conceiving of 
this relation is by assuming that work can be trans 
formed into kinetic energy and vice versa in such 
a way that given amounts of the two magnitudes are 
equal or equivalent to one another. Naturally, this 
is only an abbreviated way of expressing the actual 
relations, for it might just as well be assumed that 
the work really disappears and the kinetic energy 
really originates anew, and that the disappearance 
of the one substance only happens regularly to co 
incide with the origin of the other. But it is this 
regular conjunction of phenomena that constitutes 
the sole ground of every causal relation, and in such 
a sense we are justified in regarding the disappear 
ing work as the cause of the kinetic energy that 
arises, and to designate this relation summarily as a 
transformation. 

By the inclusion of cases in which work is con 
verted into kinetic energy the law of the conserva 
tion of work therefore becomes the law of the con 
servation of the sum of work and kinetic energy. 
We are thereby compelled to extend the concept of 
substance, which at first contains only work, to the 
sum of both magnitudes, and to introduce a new 
name for this enlarged concept. 

It will soon appear that all cases of imperfect 
machines, in which work disappears without giving 
rise to an equivalent amount of kinetic energy, can, 



Kinetic Energy 135 

with a corresponding enlargement of the concept, be 
likewise included in the law of conservation. For 
experience has shown that in such cases something 
else arises, heat, light, or electric force, etc. This 
generalized concept, which embraces all natural 
processes and permits the sum of all corresponding 
values to be expressed by a law of conservation, 
we call energy. The law in question, therefore, 
is: 

In all processes the sum of the existing energies 
remains unchanged. 

The principle of the conservation of work in per 
fect machines proves to be an ideal special instance 
of this general law. A perfect machine is one in 
which work changes into nothing but work of an 
other kind, and not into a different kind of energy. 
Then each side of the equation which expresses the 
general law of energy, namely, 

Energy that has disappeared = energy that has 
arisen, 

contains only the magnitude of the work, and ex 
presses the law of the conservation of work. If, 
on the other hand, as in the case of the pendulum, the 
work increasingly changes part by part into kinetic 
energy, and vice versa, the equation during the first 
period is : 

Work that has disappeared = kinetic energy that 
has arisen, 

and during the second period in which the pendulum 
rises again, 



136 The Physical Sciences 

Kinetic energy that has disappeared = work that 
has arisen. 

Thus, while work can be called a substance only in 
a limited sense, since its conservation is limited 
only to perfect machines, we may call energy a sub 
stance unqualifiedly, since in every instance of which 
we know the principle has been maintained that a 
quantity of any energy never disappears unless an 
equivalent quantity of another energy arises. Ac 
cordingly, this law of the conservation of energy 
must be taken as a fundamental law of the physical 
sciences. But not only do all the phenomena of 
physics, including chemistry, occur within the limits 
of the law of conservation, but until the contrary is 
proved the law of conservation must also be re 
garded as operative in all the later sciences, that is, 
in all the activities of organisms, so that all the 
; phenomena of life must also take place within the 
limits of the law of conservation. This corresponds 
to the general fact, which I have emphasized a num 
ber of times, that all the laws of a former science 
find application in all the following sciences, since 
the latter can only contain concepts which by spe 
cialization, that is, by the addition of further char 
acteristics, have sprung from the concepts of the 
former or more general sciences. 

45. Mass and Matter. It has been noted above 
that kinetic energy depends upon another magnitude 
beside velocity. A conception of its nature can be 
obtained when we try to put different bodies in mo- 



Mass and Matter 137 

tion. In doing so the muscles of the arm perform 
certain quantities of work, and we feel whether the 
quantities are greater or smaller. In this way we 
obtain a clear consciousness of the fact that differ 
ent bodies require quite different quantities of work 
for the same velocity. The property which comes 
into play here is called mass, and mass is propor 
tional to the work which the various bodies require 
to attain the same velocity. Since the work and the 
velocity can be measured very accurately by ap 
propriate means, mass also lends itself to a corre 
spondingly accurate measurement. 

All known ponderable bodies have mass. That 
means there is a regular connection between the 
property which makes a body tend to the earth with 
a certain definite force (called weight) and the prop 
erty by virtue of which a body assumes certain 
velocities under the influences of motive causes. 
We can readily conceive that it is possible for us to 
learn only of such bodies as are heavy, that is, 
bodies which are held by the earth, since the others, 
if they exist at all, would naturally have left the 
earth long ago. That all these bodies also have 
mass is to be explained in a similar \vay. For a 
body of mass zero would at each impulse assume in 
finitely great velocity, and could therefore never be 
the object of our observation. Consequently, by 
reason of the physical conditions obtaining on the 
earth s surface, the bodies known to us must com 
bine both properties, mass and weight. 



138 The Physical Sciences 

The name given to this concept of the combined 
presence of mass and weight in space is matter. 
Experience shows that there is a law of conservation 
for these magnitudes also, according to which what 
ever changes we may produce in bodies possessing 
weight and mass, no change will occur in the sum of 
their weight and mass. According to the nomen 
clature previously introduced we must therefore call 
weight and mass substances, since they remain the 
same as to quantity, no matter what changes they 
may undergo. However, it is usual to apply the 
name substance to the concept of matter composed of 
mass and weight. In fact, scientists often go so far 
as to limit the name to this single instance of the 
various laws of conservation, and to take substance 
to mean exclusively the combination of mass and 
weight. This is connected with the conception 
which we are about to discuss, that all natural 
phenomena can ultimately be conceived as the mo 
tion of matter. Through the greater part of the 
nineteenth century this conception, called scientific 
materialism, was accepted almost without opposi 
tion. At present it is being more and more recog 
nized that it was only an unproved assumption, 
which the development of science daily proves to be 
more untenable. 

46. Energetic Mechanics. In the light of our 
previous observations the branch of science tradi 
tionally known as mechanics appears as the science 
of work and of kinetic energy. Furthermore, statics 



Energetic Mechanics 139 

is shown to be the science of work, while dynamics, 
besides treating of kinetic energy in itself, also treats 
of the phenomena of the change of work into kinetic 
energy, and vice versa. We shall find the same re 
lation again later, only in more manifold forms. 
Every branch of physics proves to be the science of 
a special kind of energy, and to the knowledge of 
each kind of energy must be added the knowledge 
of the relations by which it changes to the other 
forms of energy and vice versa. It is true that in 
the traditional division of physics this system has 
not been strictly carried out, since an additional and 
very influential motive for classification has been the 
regard paid to the various human sense organs. 

Nevertheless this ground does not lie in the field 
of physics, but in that of physiology, and must, 
therefore, be abandoned in the interest of strict sys- 
tematization. 

Of the physical sciences mechanics was the first 
to develop in the course of historical evolution. A 
number of factors contributed to this end the wide 
distribution of mechanical phenomena, their sig 
nificance to human life, and the comparative sim 
plicity of the principles of mechanics, which made 
it possible to discover them at an early date. Most 
to be noted is, that of all departments of physics 
mechanics is the first which lent itself to comprehen- 
sive mathematical treatment. It is true that the 
mathematical treatment of mechanics was possible 
only after idealizing assumptions had been made 



140 The Physical Sciences 

perfect machines and the like so that the results of 
this mathematical treatment not infrequently had 
very little to do with reality. The mistake of 
losing sight of the physical problem and of making 
mechanics a chapter of mathematics has not always 
been avoided, and it is only in most recent times that 
the consciousness has again arisen that the classical 
mechanics, in arbitrarily limiting itself to extreme 
idealized cases, sometimes runs the risk of losing 
sight of the aim of science. 

47. The Mechanistic Theories. Because the evo 
lution of mechanics antedates that of the other 
branches of physics, mechanics has largely served 
as a model for the formal organization of the other 
physical sciences, just as geometry, which has been 
handed down to us from antiquity in the very 
elaborate form of Euclid, has largely been used as 
a model for scientific work in general. Such 
methods of analogy prove to be extremely useful at 
first because they serve as a guide to indicate when 
and where new sciences, in which all possibilities 
are open, can be got hold of. But later on such 
analogies are apt to be harmful. For each new sci 
ence soon requires new methods, by reason of the 
peculiar manifoldness which it has to deal with, and 
the finding and the introduction of these new 
methods are easily delayed, and, as a matter of fact, 
often have been delayed, because scientists could 
not free themselves soon enough from the old 
analogy. 



The Mechanistic Theories 141 

By its being based upon memory the human mind 
is so constructed that it cannot assimilate some 
thing entirely new. The new must in some way 
be connected with the known in order that it may be 
organically embodied in the aggregate of concepts. 
Therefore, it is the first involuntary impulse of our 
mind, in the presence of new experiences or thoughts, 
to look about for such points at which a linking of 
the unknown to the known seems possible. In the I 
case of mechanics this necessity for finding connect- I 
ing links has acted in such a way that the attempt A 
has been made, and is still being made, to conceive | 
and represent all physical phenomena as mechanical. | 

The impulse to this was first given by the extraor 
dinary successes which mechanics has attained in 
the generalization and prediction of the motions of 
the heavenly bodies. The names of Copernicus, 
Kepler, and Newton mark the individual steps in 
the mechanization of astronomy. The cause of this 
lies in the fact that the heavenly bodies actually ap 
proximate very closely the ideal of the purely 
mechanical form with which classical mechanics 
deals. These successes encourage the attempt to 
apply these mental instruments that were productive 
of such rich results to all other natural phenomena. 
An old theory, according to which all physical things 
are composed of the most minute solid particles of 
matter called atoms, supported these tendencies and 
invited the attempt to regard the little world of 
atoms as subject to the same laws as had been found 



142 The Physical Sciences 

to apply so successfully to the great world of the 
stars. 

Thus we see how this mechanistic hypothesis, the 
assumption that all natural phenomena can be re 
duced to mechanical phenomena, comes as if it 
were a self-understood matter, and with its claim to 
be a profound interpretation of nature it scarcely 
permits the question as to its justification to be 
raised at all. And the effects here have been the 
same as I described above in cases in which infer 
ences from analogy are accepted too extensively or 
too credulously. While it is true, no doubt, that 
the mechanical hypothesis at first was fruitful of re 
sults in special research, because it facilitated the 
putting of the question for example, we need 
think only of the atomic hypothesis in chemistry- 
later, the efforts to find further hypothetic help for 
the inadequacies of the hypothesis that gradually 
came to light, have not infrequently led scientific re 
search to pseudo-problems, that is, to questions which 
are questions only in hypothesis, but to which no 
actual reality can be shown to correspond. Such 
problems, therefore, are by their very nature in 
soluble, and constitute an inexhaustible source of dif 
ferences of scientific opinion. 

The most flagrant of the injurious consequences 
i of the mechanistic hypothesis appear in the scientific 
treatment of the mental phenomena. Ready as sci 
entists were to represent all other life phenomena, 
such as digestion, assimilation, and even generation 



The Mechanistic Theories 143 

and propagation, as the consequence of an extremely 
complicated play of certain atoms, their courage 
never went so far as to apply this principle to men 
tal life and to consider that by mechanics the last 
word had been said on the subject. 

It is because of this hesitancy to bring mental 
phenomena under the same mechanistic principle as 
all the other phenomena that the philosophical sys 
tems had to search for some other means to con 
nect the mental world with the mechanical, and the 
efforts of the philosophers to bring about this end 
have been most varied. Of the various doctrines 
that have come down to us, that of the pre-estab 
lished harmony proposed by Leibnitz is in the 
ascendant in our day, and is now called the theory 
of the psycho-physical parallelism. According to 
this theory it is assumed that the mental world ex 
ists alongside, and quite independent of, the mechan 
ical, but that the things have been so prearranged 
that mental processes take place simultaneously with 
certain mechanical processes (according to some, 
with all mechanical processes) in such a way that, 
although the two series do not influence each other 
in the least, they always correspond to each other 
precisely. How such a relation has come about and 
how it is maintained remains unsaid, or is left to 
future explanation. 

We need only think of the content of this hypoth 
esis with an unbiased mind to lose all relish for it 
at once. In fact, it has no other raison d etre than 



144 The Physical Sciences 

the presumption that the mental and the mechanical 
world are opposed to each other. As soon as we 
abandon the thesis that the non-mental world is ex 
clusively mechanical, we acquire the possibility again 
of finding for the theory of mental phenomena a 
constant and regular connection with the theories 
of all other phenomena, especially with the phenom 
ena of life. Therefore it will be found most ex 
pedient in every respect, instead of rendering sci 
entific research one-sided and almost blind to non- 
conforming facts by preconceived hypotheses, such 
as the mechanistic hypothesis, to seek, as hitherto, 

, from step to step, the new elements of manifoldness 
which must be taken account of in the progressive 
upbuilding of science and to limit ourselves faith 
fully to them in the formation of general ideas. 

48. Complementary Branches of Mechanics. The 
field of pure or classical mechanics is limited to the 
above two kinds of energy, work and kinetic energy, 

! though these do not exhaust the manifoldness of the 
mechanical energies. Accordingly, other branches 
of mechanics dealing with the corresponding phe 
nomena are added to the classical mechanics de 
scribed above. 

If by mechanical energies we understand all 
energies in which changes of space are connected 
with changes of energy, there are as many differ 
ent forms as there are spacial concepts that seem ap 
plicable. Form, Volume, and Surface of bodies 
in space are especially recognizable as the field of 



Complementary Branches of Mechanics 145 

action for energy, which shows different properties 
or manifoldnesses according to each of these re 
lations. 

The energy of form is manifested in bodies (solid 
or rigid bodies) that maintain a definite shape be 
cause every change of shape is connected with work 
or with the expenditure of some other energy. If 
the changes are small, the bodies are of such a na 
ture that they return to their former condition of 
their own accord after the force exerted upon them 
has ceased to act. This property is called elasticity. 
However, the theory of elasticity, which has been 
extensively and rationally developed, is regarded as 
belonging rather to mathematical physics in gen 
eral than to mechanics in particular. In greater 
changes of shape the energy of form, or elastic 
energy, passes into other forms, and the body does 
not return to its former shape after the force has 
been removed. 

Other bodies have no energy of form (or only 
in an inrmitesimally slight degree), so that they al 
low of changes of form without the expenditure of 
work, but their volume can be changed only by 
work. These are divided into two classes. First, 
the liquids, which have a definite volume (corre 
sponding to the definite shape of solids), the changes 
of which in every sense, both compression and ex 
pansion, require work. Secondly, the gases with 
volume energy in only one sense of the \vord, in 
which only the compression of volume requires 



146 The Physical Sciences 

work, while in expansion a certain amount of work 
is thrown off. Such bodies can exist only so long 
as the expenditure of their volume energy by spon 
taneous expansion is prevented by the presence of 
a counter energy, as, for example, the elasticity 
of the walls of a vessel. This tendency is called 
pressure. 

Finally, there are energy qualities at the surfaces 
between various kinds of bodies which come into 
play at the change of these surfaces. They always 
lie in such a direction that the enlargement of the 
surfaces requires work, and hence, by reason of the 
law of conservation of energy, cannot proceed by 
itself. In cases where there has been an inverse 
kind of energy present, that is, one which diminishes 
with increasing surface, it also has been active as 
a rule, thus bringing about the disappearance of the 
existing boundaries. 

Since the seat of this kind of energy is in the 
surfaces (or superficies), it is called surface-energy. 
The phenomena depending upon it manifest them 
selves most clearly at the surface boundaries be 
tween liquids and gases. They are called capillary 
phenomena. This strange name, derived from the 
word capilla, hair, has its origin in the fact that 
because of surface-energy liquids rise in tubes which 
they wet, and the narrower the tube the higher they 
rise. If the lumen of the tube is as fine as a hair, 
a considerable rise can be observed. This is the en 
tire connection between the name and the thing. 



The Theory of Heat 147 

The mechanics of liquids is called hydromechan 
ics, that of gases, aeromechanics, after the most 
familiar liquid, water, and the most familiar gas, 
air. The study of surface-energy under the name 
of the capillary theory forms part of theoretical 
physics. While formerly this branch, too, was re 
garded as a working part, or, rather, as a playing 
part, of mathematical problems, in more recent 
times extensive experimental research has made its 
entry in this province also, and has demonstrated 
the necessity of passing from the former abstrac 
tions or idealizations, which were carried altogether 
too far, to a better and pro founder regard for the 
actually existing complexities. 

49. The Theory of Heat. The various forms of 
energies the aggregate of which is comprehended in 
physics, have very different special characters. A 
systematic investigation has not yet been made of 
the characters of manifoldness by which, for ex 
ample, work is distinguished from heat, electrical 
energy from kinetic energy, etc., nor of what are the 
essential properties peculiar to each individual 
energy. We feel certain that differences do exist, 
for otherwise the energies could not be distin 
guished, and we feel certain that these differences 
are very important, for doubt seldom arises as to the 
kind of energy to which a certain phenomenon is 
to be assigned. But just as we have no systematic 
table of the elementary concepts, so we are still with 
out a systematic natural history of the forms of 



148 The Physical Sciences 

energy in which the peculiarities of every species 
are characterized, and in which the entire material 
is so arranged according to these characteristics that 
we can take a general survey of it. 

As regards heat energy, its foremost and most 
striking characteristic is its physiological effect. In 
our skin there are organs for the perception of heat 
as well as of cold, that is, for temperatures above 
and below the temperature of the skin. However, 
the temperature that these organs can bear without 
injury to themselves is of a very small range, be 
yond which physical apparatuses of all kinds must 
be used, such as " thermometers." 

Heat is the simplest kind of energy from the 
point of view of manifoldness. Every heat quan 
tity is marked by a temperature, just as a kinetic 
energy is marked by velocity. But while a velocity 
is determined in space so that velocities of equal 
magnitude have in addition a threefold infinite mani 
foldness in reference to direction, a temperature is 
characterized completely and unambiguously by a 
simple number, the degree of temperature. Two 
temperatures of equal degree can in no wise be dis 
tinguished, since temperature possesses no other 
possible manifoldness than degree. 

The same property is found in heat energy itself. 
In heat energy we measure the quantity of energy 
itself and call it the heat quantity, while in some of 
the other kinds of energy, only the factors into 
which they can be divided are measured, and no 



The Theory of Heat 149 

habitual conception of the energy itself is developed. 
A heat quantity is likewise fully indicated by its 
measure number. 

That heat is an energy, that is, that it is developed 
in equal quantities from other kinds of energy, and 
can change back again into them, is a discovery 
which, despite its fundamental and general char 
acter, was not made before the forties of the nine 
teenth century. As often happens in cases of im 
portant scientific advances, the same idea came simul,- 
taneously to a number of investigators. The first 
to grasp and fully comprehend this idea was Julius 
Robert Mayer of Heilbronn, who published his re 
sults in 1842. Mayer not only showed that the 
imperfect machines (p. 134), which limit the validity 
of the law of the conservation of work, owe this 
peculiarity to the fact that they transform a part 
of the work into heat, and that when we take account 
of this part, the law of conservation holds perfectly 
good, but he also calculated, with extraordinary 
acumen, the mechanical equivalent of heat from the 
then existing data of physics. That is to say, he 
determined how many units of heat (in the measure 
then in use) correspond to a unit of work (in its 
specific measure) in the change from one to the 
other, and back. And this fundamental knowledge 
of the existence of a quantitatively unchangeable 
substance, arising from work, and capable of being 
transformed into it, Mayer did not limit in its ap 
plication merely to heat. He was the first to con- 



150 The Physical Sciences 

struct a table, which he made as complete as pos 
sible, of all the forms of energy then known, and 
to assert and prove the possibility of their reciprocal 
change into each other. 

In view of this relation of the quantitative equiv 
alent of the various forms of energy when trans 
formed into one another, an attempt is being made 
at present to measure them all with the same unit. 
That is, some easily obtained quantity of energy is 
arbitrarily chosen as a unit and it is determined 
that in every other form of energy the unit shall 
be equal to the quantity obtained from that unit on 
its transformation into the energy in question. For 
formal reasons the kinetic energy of a mass of two 
grams which moves with the velocity of one centi 
meter in a second has been chosen as the unit. It 
is called erg, an abbreviation of energy. The 
amount is very small, and for technical reasons io 10 
times greater unit is used. To raise the tempera 
ture of a gram of water one degree a quantity of 
energy equal to 41,830,000 ergs is required. 

50. The Second Fundamental Principle. An 
other fundamental discovery has been made in con 
nection with the heat form of energy, which, like 
the law of conservation, relates to all forms of 
energy, but has found its first and most important 
application in heat. While the law of conserva 
tion answers the question, how much of the new 
form of energy is developed if a given quantity of 
energy changes, but gives no clue as to when such 



The Second Fundamental Principle 151 

a change occurs, this second law asserts the condi 
tion under which such changes arise, and is there 
fore called the second fundamental principle. 

The discovery of this law antedates Mayer s dis 
covery of the law of conservation by about twenty 
years, and was made by a French military engineer, 
Sadi Carnot, who died soon afterward without hav 
ing lived to see the recognition his great work ob 
tained. Carnot asked himself the question, Upon 
what does the action of the steam engine, which had 
just then come into use, depend? This led him 
first to the more general question of the action of 
heat engines in general. He found that no heat 
engine could work unless the heat dropped from a 

higher to a lower temperature, just as no water 
wheel can work unless the water flows from a 
higher to a lower level, and he determined the con 
ditions which an ideal heat engine must fulfil, that 
is, a machine in which the greatest possible value in 
work is obtained from heat. However, an ideal 
machine of this nature can be constructed in very 
different ways, and Carnot s discovery consists in 
the recognition of the fact that the quantity of work 
obtained from the heat unit does not at all depend 
upon the peculiar construction of the ideal machine, 
but is determined solely by the temperature between 

\ which the heat transition takes place. This fol 
lows from the following considerations : 

In the first place an ideal engine must be re 
versible, that is, it must be capable of working both 



152 The Physical Sciences 

ways, changing heat into work and work back into 
heat. Now, if we have two ideal engines between 
the same temperatures, and if we assume that en 
gine A produces more work from the same quan 
tity of heat than engine B, then let A move one 
way and let B move the other way with the work 






obtained from A. Since B produces less work from 
a given amount of heat, hence more heat from an 
equal amount of work, there will in the end be 
more heat at the higher temperature than was orig 
inally there. But experience teaches that there is 
no means in nature by which heat in the absence of 
concomitant change could be caused to rise to a 
higher temperature. Therefore an engine so con 
structed as to produce this result is impossible, And 
B cannot be of such a nature as to produce less 
work from the same quantity of heat than A. 

The reverse is also impossible. For then we 
need merely couple the engines in the reverse way 
in order to obtain the same effect. Therefore, since 
B can do neither less nor more work than A, the 
two must do the same amount of work which was 
to be proved. 

It is obvious that this process of proof is similar 
to that by which the law of conservation was estab 
lished. Because the arbitrary creation of energy 
from nothing is impossible there must be definite 
and immutable relations of change between the 
forms of energy. Because energy at rest does not 
spontaneously pass into conditions in which it can 



The Second Fundamental Principle 153 

do work, the efficiencies of the machines must have 
definite and unchangeable values. If, for example, 
we could cause heat of its own accord to rise to a 
higher temperature, we could also construct a per 
petual motion machine which would always yield 
work at no expense. But this perpetual motion 
would not be one that creates work out of nothing, 
but one that extracts it from energy at rest. A per 
petual motion machine of this nature, too, is, accord 
ing to our experience, impossible, and this impos 
sibility forms the content of the second fundamental 
principle. 

On the face of it this apparently " self-evident " 
proposition does not reveal how fruitful of results 
it is when applied to the discovery of simple but 
not obvious relations. It can only be said here that 
the deductions from this principle form the chief 
content of the extensive science of thermodynamics, 
which deals with the changes of heat into other 
forms of energy. We must only emphasize the 
fact that the application of this law, as was already 
observed in stating it, is not confined to the changes 
of heat alone. It is a law rather which finds ap 
plication in all the forms of energy. For in every 
form of energy there is a property which corre 
sponds to temperature in heat, and upon the equality 
or the inequality of which depends whether the 
f energy in question is at rest or ready for trans 
formations. This property is called the intensity 
of the energy. In work, for instance, it is force, 



154 The Physical Sciences 

in volume-energy it is pressure. If once the in 
tensity in a body is equal, its energy is at rest, and 
it never again moves of its own accord. 

Another form in which to present these relations 
is to make a distinction betweenjfrgg energy and 
energy at rest. If we have a heat quantity the tem 
perature of which is higher than that of the sur 
rounding objects, it can be used to do work only 
* until its temperature has dropped to that of the sur 
rounding objects. Although energy in abundance 
is still present, there is no longer any energy capable 
of change, or free energy. Since differences of 
temperature, like other differences of intensity, have 
a constant tendency to diminish, the amount of free 
energy on earth is constantly decreasing, and yet 
it is only this free energy that has value. For since 
all phenomena depend upon change of energy, and 
change of energy is possible only through free 
energy, free energy is the condition of all phe 
nomena. 

51. Electricity and Magnetism. While the 
knowledge of heat energy goes back to the most 
ancient periods of civilization, electrical and mag 
netic energies are relatively young acquisitions. 
The highly developed technical application of both 
with the rich harvests they have yielded belongs ex 
clusively to most recent times. 

Both these forms of energy, like those discussed 

above, are connected in the main with ponderable 

1 " matter," but in a much slighter and less regular 



Electricity and Magnetism 155 

measure. While it is not possible as yet to ren 
der any given body free of heat (although lately 
the absolute zero point has been considerably ap 
proximated), freedom from electrical and magnetic 
energy is the normal condition of most bodies. This 
is connected with the peculiarity that electrical and 
magnetic properties are decidedly bi-symmetrical or 
polar. This property is not found in any other form 
of energy, and can serve as the special scientific 
characteristic of electricity and magnetism. This 
peculiarity shows itself in the concepts of positive 
and negative magnetism, and positive and negative 
electricity, and is due to the fact that two equal op 
posite quantities of electricity or magnetism, when 
added together, do not produce double their value, 
but nullify each other.* 

The fact that electrical and magnetic energies 
generally exist only in a transitory state (with the 
notable exception of the magnetic condition of the 
earth) is probably the cause of our not having de 
veloped a sense organ for them, especially since 
their phenomena as they occur in nature have only 

* For the sake of the layman it must be observed that those 
" quantities " are not energy magnitudes but factors of the 
electrical and magnetic energies. Energy itself in its various 
forms is an exclusively positive magnitude, and the result of 
the additions of their various amounts is always the sum, 
never the difference, of their numerical values. By the 
negative sign is understood the energy expended in contra 
distinction to the energy received. It is therefore nothing 
more than the indication of a mathematical operation. 



156 The Physical Sciences 

occasionally and in very rare instances (thunder 
storms) an influence upon us. On the other hand, 
the modern development of electrotechnics is based 
upon that property of electrical energy by virtue of 
which large quantities of it can be conducted along 
a thin wire over great distances without any con- 

*siderable loss, and at the point desired can be easily 
changed into any other forms of energy. But since 
the collection and conservation of large quantities 
of electrical energy is hardly possible technically, 
the electrical apparatus must be so constructed that 
the quantities each time required should be pro 
duced at the moment they are used. The chief 
source of electricity is the chemical energy of coal, 
which is first transformed into heat, then into 
mechanical energy, and finally into electrical energy. 
This extremely roundabout process is necessary be 
cause a method technically practicable of transform 
ing the chemical energy of coal directly into elec 
trical energy has not yet been invented. On the 
other hand, mechanical energy can be easily and 
completely changed into electrical energy. Upon 
this is based the exploitation of much " water 
power," the energy of which could not be utilized 
but for the great capacity for change of the elec 
trical form. 

52. Light. The case of light in our day seems to 
be similar to that of sound, which, although it has 

jits special sense organ in man, is yet no particular 
form of energy, but has been found to be a com- 



Light 157 

bination of mechanical energies in an oscillatory or 
mutually changing state. It seems highly probable 
that light, too, is not a special form of energy, but 

/ a peculiar oscillatory combination of electrical and 
magnetic energies. It is true that the circle of proof 
is not yet quite closed, but the gaps have become so 
small that the above conclusion may at any rate be 
accepted as highly probable. 

However that may be, light is an energy which, 
according to the known laws,, travels through space 
with tremendous rapidity. We will call it radiant 
energy, since the part optically visible, to which alone 
the name light in its original sense belongs, repre 
sents an extremely small portion of a vast field, the 
properties of which change quite continuously from 
one end to the other. 

Radiant energy is characterized as an oscillatory 
or wave-like process. So long as this fact was un 
known (up to the beginning of the nineteenth cen 
tury) it was thought that light consisted of minute 

+ spherical particles, which shot through space in a 
straight line with the tremendous velocity mentioned 
above. Later, in order to " explain " its wave na 
ture, which in the meantime has come to be recog 
nized, it was assumed to be due to the elastic vibra- 

* tions of an all-pervading thing called ether, of which 
we know nothing else. This elastic undulatory 
theory has been abandoned in our time in favor 

of an electromagnetic theory supported by quite 
considerable experiential grounds. Whether it 



158 The Physical Sciences 

will be spared the fate that has overtaken the 
older theories (or rather hypotheses) of light 
cannot as yet be predicted with any degree of 
certainty. 

Radiant energy is of very marked importance in 
human relations. As light it serves, with the aid of 
the corresponding receiving organs, the eyes, as a 
more manifold means of intercommunication be 
tween our bodies and the outer world than any other 
form of energy. The energy quantities penetrat 
ing to us from the extreme limits of the world 
space mark the outermost limits of which we have 
knowledge in any way whatsoever, and finally the 
energy quantities radiating to us from the sun con- 
; stitute the supply of free energy at the expense of 
which all organic life on earth is maintained. Even 
the chemical energy stored up in coal repre 
sents nothing else than accumulations of former 
sun radiation, which had been transformed by 
the plants into the permanent form of chemical 
energy. 

Very recently other newly discovered forms of 
radiant energy have been added to light. They are 
produced in manifold circumstances, and some 
bodies emit them constantly. The scientific elabora 
tion of these extremely manifold and unusual phe 
nomena has not yet been carried so far that they 
can be reduced to a doubt-free system. But so 
much, it seems, is already apparent, that they are 
presumably not purely new forms of energy, but 



Chemical Energy 159 

rather very composite phenomena which may yield 
one or more new energies as component parts. But 
despite the peculiarity of these new rays, nothing 
certain has as yet been proved against the law of 
conservation itself. 

53. Chemical Energy. Since chemical energy is 
only one of several forms of energy, there seems to 
be no justification for allotting it to a special science, 
1 since all the other forms of energy must be incor 
porated in physics. 

But the actual existence of chemistry as a spe 
cial science which has already many subdivisions is 
justified in the first place by the external fact that 
in practical life and in industry chemistry occupies 
a very wide field comparable, if not superior, to 
that of the whole of physics. In the next place, 
from the psychological point of view, it is found 
that the chemist s methods of reasoning and work 
ing are so different from those of the physicist that 
a division seems to be in order for that reason also. 
Finally, there is in the nature of chemical energy 
itself an important distinction which marks it off 
from the other forms. 

While, for example, there is only one form of 
heat or of kinetic energy, and in electricity there 
are only the two forms of polar opposites, chemistry, 
even after the greatest theoretical reduction, pos 
sesses at least about eighty forms. That is, it pos- 
sesses as many forms as there are chemical elements. 
The experiential law, that the elements cannot be 



160 The Physical Sciences 

changed into one another,* also limits the corre 
sponding changes of the chemical energies into one 
another, and thus characterizes the independence of 
these various forms. From this results a dispropor 
tionately greater manifoldness of relations, which 
find their expression in the many thousands of the 
individualized chemical substances or combinations. 

This great manifoldness and the slight regularity 
hitherto found in connection with the properties and 
reciprocal relations of the numerous chemical ele 
ments renders modern chemistry more a descriptive 
than a rational science. It was no more than twenty 
years ago that an earnest and successful attempt 
was begun to apply the stricter methods of physics 
to the investigation of chemical phenomena. These 
labors, so far as they have gone, have yielded a 
great many far-reaching and comprehensive prin 
ciples. 

The significance of chemistry in human life is 
twofold. In the first place the energy of the human 
body, just as that of all other living organisms, de- 
fpends chiefly upon the action of chemical energies in 
the most manifold forms. Of all the physical sci 
ences, therefore, chemistry is the most important 
for biology, particularly for physiology. In the sec 
ond place, as I have emphasized a number of times, 

* Lately changes of elements into one another have been 
observed in individual instances, but in such peculiar circum 
stances that for the present we need not consider these dis 
coveries, which have only just begun. 



Chemical Energy 161 

it possesses the peculiar property which enables it to 

^be preserved for a long time without passing into 
other forms and being dissipated. Furthermore, 
energy in this form permits of the most powerful 
concentration. More of chemical energy can be 

stored in a given space than of any other form of 
energy. Both these properties, then, may be con 
sidered as the reason why organic beings are con 
stituted chiefly by means of chemical energy. At 
any rate, it is due to these two peculiarities that 
chemical energy serves as the primary source for 
almost all the energy used in industry. 

Further, the manifoldness of chemical energy is 
the cause of the peculiar manner in which it is 
transformed into other forms. In the other forms 
of energy the transformation can be effected by the 
body itself. Nothing else is required. If a stone 
is thrown and it hits against a wall, it loses its 
kinetic energy, the greater part of which changes 
into heat. But in order to liberate the chemical 
energy of, say, coal, the coal alone is not sufficient ; 

1 another chemical substance is required, the oxy 
gen of the air. The interaction of the two sub 
stances produces a new substance, and it is only 
during this process that a corresponding part of 
the chemical energy is liberated. There are a few 
chemical processes also (allotropic and isomeric 

changes) in which a single substance without the 
co-agency of another substance can give off energy. 
But the quantity of energy thus obtained is in- 



162 The Physical Sciences 

finitely small as compared to that liberated by the in 
teraction of two or more substances. Because of 
the necessity of two or more substances to co-operate 
in giving off chemical energy, the opportunity for 
the transformation of chemical energy is less than 
for the transformation of the other forms of energy, 
and this is the main reason why it can be conserved 
so long and so easily. All that is necessary is to 
prevent contact with another substance. This is a 
problem, it is true, which from the point of view of 
strict theoretical rigor it is almost impossible to 
solve. In practice, however, it can be easily solved 
for periods of time long enough at least to require 
special means to enable us to recognize that it is only 
a temporary and not a fundamental solution. Sci 
entifically expressed, the cause of this is that the dif 
fusion of the various substances in one another can 
theoretically never be completely eliminated, while 
on the other hand the velocity of the diffusion over 
distances measured only by decimeters is extremely 
low. 



PART IV 
THE BIOLOGIC SCIENCES 

54. Life. Among the bodies in our environment 
that are ponderable and have mass the animate be 
ings are so strikingly distinguished from the in 
animate that in most cases we have not the slight 
est doubt whether a body belongs to the one kind or 
to the other, even if in some cases we happen not 
to be familiar with its peculiar form. In the first 
place, therefore, we must answer the question in a 
general way and tell what the distinguishing peculi 
arities are that mark them off one from the other. 

The first peculiarity is this, that living organisms 
are not stable but stationary forms. This distinc 
tion is based upon the fact that a stable form is at 
rest or unchangeable in all its parts, while a sta 
tionary body, though it seems unchangeable in its 
form, internally undergoes a constant change of its 
parts. Thus, a brass faucet is a stable body, since 
it not only preserves its form and function perma 
nently, but consists at all times of the same material 
and shows the same peculiarities, such as stains, de 
fects in form, etc. It cannot be said, it is true, 
that it will remain completely unchanged for all 

163 



164 The Biologic Sciences 

time. Its metal suffers a gradual chemical and 
mechanical deterioration. But this is not essential 
to the existence of the faucet, since the deterioration 
varies greatly with circumstances, and if conditions 
are ideal it can be reduced to zero. 

On the other hand, the jet of water flowing from 
the faucet is a stationary body. In favorable cir 
cumstances it can assume a constant form, so that 
at a hasty glance it might be taken for a stable glass 
rod. On closer examination it will be found that 
the parts of water of which it is formed are not the 
same at any given instant as the instant before, each 
part that has flowed away being replaced by another 
just as large following it. 

From this difference in the nature of the two 
bodies results a difference in their behavior. If I 
make a mark on the faucet with a file, the mark re 
mains permanent. But even if I sever the entire 
water jet with a knife, the cut is healed the next 
moment, because by reason of the continuous flow of 
the water, the severed place is instantly eliminated 
from the body. Owing to this nature peculiar to 
^stationary bodies, they have the capacity of being 
healed or of regeneration. 

For a body to continue permanently in a stationary 
condition the material of which it is composed must 
be permanently supplied. If we turn off the faucet, 
the water jet immediately disappears or " dies." 
Evidently, therefore, a stationary body can subsist 
by its own means only if it has the property or 



Life 165 

capacity to provide itself continually with the neces- 

* sary material. This material consists in the main 
of ponderable or chemical substances of definite 
physical and chemical properties, and thus the change 
of substance, metabolism, appears as a necessary 
property of the stationary body. In order, how 
ever, that metabolism should take place we must 

have free energy, or energy having the capacity to 
work, since it is only free energy that can cause sub 
stances to change, just as every phenomenon in the 
world implies the equalization of free energy. For 
a stationary body to exist independently, therefore, 
it must have the property of being able spontane 
ously to possess itself of the necessary substances 
and of free energy. But since, as we have already 
said, the energy of organisms is stored up and used 
in the main in the form of chemical energy, the two 
tasks which a stationary body has to perform, that 

of meeting the need for substances and for energy, 
are as a rule externally combined. In organisms 
these two necessities combined are called nutrition, 
and thus we recognize in the capacity for self- 

* acquisition of nutrition another essential property 
of organisms. 

A third essential property of organisms is the 
capacity for reproduction, for the bringing forth of 
similar beings. It is never impossible that the bal 
ance between the receipts and expenditures of a 
stationary body should, in consequence of some ex 
ternal causes, be disturbed, even when under nor- 



1 66 The Biologic Sciences 

mal conditions it possesses the property of self- 
nutrition. If the disturbance remains below a cer 
tain point, then, as we have already stated, regenera 
tion sets in. But the disturbance may rise above 

that point, in which case the body ceases to exist, 
or dies. Then a similar body will not arise unless 
the manifold necessities that have led to the origin 
of the first will combine again to produce the sec 
ond. That such a thing is possible, that, in fact, 
it often happens, is shown, for example, by the 
waves of the ocean, which have a stationary char 
acter since, while they are composed of constantly 
changing masses of water, their form remains un 
changed. The waves are destroyed in the breakers, 
but arise again and again through the action of the 
wind upon the surface of the water. But the more 
complex such bodies are the less easily they are 
formed, while once they have been formed and have 

*found the conditions of their existence, their pres 
ervation is much easier. 

Beings, therefore, which have the capacity to 
form similar bodies out of themselves regularly and 
at the right time can preserve their species much 
more easily than those in which this property is ab 
sent. Death has to a great extent lost its power 
over beings capable of reproduction. By way of 
illustration let us take another stationary thing, a 
flame. A flame is not an organism because it is 
not self-sustaining. Yet it multiplies itself. And 
while a single little flame soon dies out, the sea of 



Life 167 

flame of a burning forest, which started from a 
single small flame, is well-nigh inextinguishable, 
and it cannot be fought in any other way than by 
letting it die its natural death and burn to the end. 

Thus, while the fulfilment of the first two condi 
tions, the stationary change and the self -supply of 
food, could produce bodies, which would be able to 
exist for a longer or shorter period, but which at 
some time would have to give way to other bodies 
of different form and nature, the capacity for repro 
duction creates the condition that forms of the same 
G species continue to exist even after the existence of 
the individual has ceased. 

These three properties constitute the essential 
characteristics of animate things or organisms. 

That the organisms are all constructed upon the 
.basis of chemical energy is a fact of experience 
which may be understood to imply that the other 
forms of energy are not capable of producing the 
above-mentioned conditions. This is due to the 
properties of chemical energy to which I have al 
ready called attention : its great concentration and, 
* at the same time, its capacity for prolonged pres 
ervation. That chemical energy is the only form 
, of energy suitable to life is obvious from the fact 
that in airship navigation, for example, the kinetic 
energy required for steering can be supplied only in 
the form of gasoline or hydrogen, that is, in the 
form of chemical energy, because any of the other 
forms would be much too heavy. The flight of a 



1 68 The Biologic Sciences 

bee or the swimming of a dolphin cannot be con 
ceived of except as brought about through chemical 
energy. 

That this chemical energy is essentially that of 
carbon has also been established by experience, al 
though it is not quite universal, for the sulphur 
bacteria found their household upon the energy of 
sulphur. The cause of the preference of carbon 
is again to be sought in its special fitness for the 
purpose, due, on the one hand, to its wide distribu 
tion, and, on the other hand, to the exceeding mani- 
foldness of its combinations. 

Finally, the construction of the organisms from a 
( peculiar combination of solid and liquid substances 
can be proved to be equally due to technical rela 
tions. 

These three last-named peculiarities are therefore 
to be regarded as the special characteristics of the 
organisms with which we are acquainted on the sur 
face of the earth in the conditions there prevailing. 
I We need not regard them conceptually as unchange 
able or irreplaceable. But the first three character 
istics, namely, the stationary nature, self-supply of 
nutrition, and reproduction, we may regard as the 
essential characteristics of organisms. They con 
stitute the frame within which everything must be 
found which we should recognize as living in the 
widest sense. 

55. The Storehouse of Free Energy. If we ask 
whence the organisms obtain the free energy which 



The Storehouse of Free Energy 169 

they require for the maintenance of their stationary 
existence, the answer is that solar radiation alone 
furnishes this supply. Without this permanent sup 
ply the free energies upon the earth, so far as our 
knowledge goes, would long ago have reached a 
state of equilibrium, and the earth s bodies would 
be stable, that is, dead and not stationary and 
living. 

It is comprehensible, therefore, that machines 
should have evolved in the organism for trans form- 
% ing the radiant energy of the sun- into a permanent 
form, and, as we have already learned, chemical 
energy is permanent, while radiant energy is an ex 
tremely transitory form of energy, that is, it 
changes very readily. The very fact that, owing to 
the change from day to night, the supply of radiant 
energy periodically ceases, makes the storing-up of 
energy for the night necessary to the existence of a 
form dependent upon it. Thus, we recognize in 
the photochemical processes, that is, in the trans 
formation of radiant energy into chemical energy, 
the foundation of life on earth. 

This work is done by the plants, which thus pro 
vide a store of free energy not only for their own 
needs but also for all the other organisms which 
possess themselves directly or indirectly of the 
plant-chemical supplies in order to utilize them for 
their individual purposes. In this manner nourish 
ment in the widest sense is secured for all organ 
isms, being based upon the regular supply of free 



170 The Biologic Sciences 

energy derived from the sun. This also explains 
^the great chemical similarity of all organisms, which 
could not subsist if they were not so constructed 
as to be able to utilize the chemical energy in the 
form in which it is provided by the plants. 

Of the great stream of free energy poured out 
from the sun into cosmic space the earth receives 
an extremely small portion (corresponding to the 
bit of space it occupies in the heavenly sphere as 
seen from the sun), and the plants collect and 
store up only a very small fraction of this portion 
received by the earth. Measurements have shown 
that in most favorable circumstances a plant 
leaf changes only about 1/50 of the radiant energy 
it receives into chemical energy. If we consider 
that only a small part of the surface of the earth is 
covered with plants and that during the winter no 
energy from the sun is stored up at all, we per- 
ceive what infinite possibilities for development there 
still are in arresting and storing up free energy. 
The part stored up by the plants flows from these 
into the countless streams, brooks, and veins of the 
other organisms, to end finally as used-up energy, or 
energy at rest. This energy is at rest, it is true, 
only in relation to the earth s surface. We do not 
know whether the radiation from the earth, which 
at present amounts to about as much as the radia 
tion from the sun to the earth, is in its turn some 
where utilized. 

While the free energy is poured out in such a 






The Soul 171 

stream in one direction, the ponderable substances 
^of which the organisms are made up circulate 

through plants and animals and back again. This 

is especially true of carbon, which is freed from its 
" combination with oxygen, that is, from carbonic 

acid, by the sun energy transformed in the plants. 

While carbon serves to build up the plant body and 

represents its supply of chemical energy, the oxygen 
is returned to the air. These two substances are 
again chemically combined in the various organisms 

- and the quantities of energy which were necessary 
for their decomposition are again available for the 
manifold functions of life. The product of the 
chemical combination, carbonic acid, returns to the 
air and is ready for renewed decomposition in the 
plants. 

Thus, the entire mechanism of life can be com- 
pared to a water-wheel. The free energy corre 
sponds to the water, which must flow in one direc 
tion through the wheel in order to provide it with 
the necessary amount of work. The chemical ele- 
. ments of the organisms correspond to the wheel, 
which constantly turns in a circle as it transfers the 
energy of the falling water to the individual parts 
of the machine. 

56. The Soul. Our observations so far have 
shown the organisms to be extremely specialized in 
dividual instances of physico-chemical machines. 
Now we have to take into consideration a property 
which seems markedly to distinguish them from 



172 The Biologic Sciences 

the lifeless machines, and which we have already 
encountered in the very beginning of our treatise. 

It is the property which we there called memory, 
and which we will define in a very general way as 
I the quality by virtue of which the repetition in or- 
/ ganisms of a process which has taken place a num 
ber of times is preferred to new processes, because 
it originates more easily and proceeds more 
smoothly. It is readily apparent that by this prop 
erty the organisms are enabled to travel on the sea 
of physical possibilities as if equipped with a keel, 
by which the voyage is made stable and the keeping 
of the course is assured. 

If we ask whether this is exclusively a quality of 
organisms the question cannot be answered af 
firmatively. Inanimate bodies also have something 
like the quality of adaptation. An accurate clock 
attains its valuable qualities only after it has been 
going for some time, and the best violin is " raw " 
until it has been " broken in." An accumulator 
must be " formed " before it can do its normal 
amount of work. All these processes are due 
to the fact that the repetition of the same process 
improves the action, that is, it facilitates or in 
creases it. 

Adaptation or memory, then, is not limited to 
organisms. In inanimate things, however, this 
property is comparatively rare. Memory, there 
fore, is to be regarded as another property of organ 
isms representing an extreme specialization of the 



The Soul 173 

inorganic possibilities. This is an important point 
of view for what follows. 

In the first place, this property of adaptation facil 
itates and assures nourishment. If we take the fun 
damental idea developed by Darwin, that that pre 
dominates in the world which by virtue of its prop 
erties endures the longest time, then it is evident 
that a body which teleologically preserves and 
elaborates its nourishment will live longer than a 
similar body without this property. Moreover, by the 
general process of adaptation, these " teleological " 
f properties come to be more greatly developed and 
more readily exercised in the body that lives longer, 
so that its long life gives it another advantage over 
its rival. Thus we can understand how this prop 
erty of adaptation, which at first is to be conceived 
.of as a purely physico-chemical quality is found 
developed in all organisms. 

In its most primitive forms the quality of adapta 
tion gives rise to the phenomena of reaction, or to 
" reflex actions, that is, to a series of processes in 
the organism in response to the stimulus of an out 
ward energy. This response is made in further 
ance of the life of the organism. Reactions that 
serve a certain end, that is, teleological reactions, can 
naturally be developed to such stimuli alone to which 
the organism is frequently and regularly subjected. 
This is why adaptation to unusual phenomena is 
generally lacking, and in relation to them the or 
ganisms are often extremely unfit. The typical ex- 



174 The Biologic Sciences 

ample of this is the moth, which flies into the light 
and is burned. 

As the reactions become more fixed they develop 
into longer and more complicated series, which then 
appear to us as Instinctive actions. But here, too, we 
find the characteristic unsuitability when unwonted 
circumstances arise, even if the teleologic reactions 

\to stimuli become more manifold. 
Finally, there are the conscious acts which ap 
pear to us to be the highest degree of the series. It 
is with the teleologic regulation of these conscious 

.acts, including the very highest activities of man 
kind, that this book deals. They are distinguished 
from instinctive action by the fact that they no 
longer proceed in a single and definite series, but are 

combined at need in the most manifold ways. But 
the fundamental fact, namely, that actions are based 
upon the repetition of coinciding experiences, at 
once appears here also, since the basis of the entire 
conscious life of the soul, the formation of con- 

- cepts, is made possible only through repetition. 
Thus, we are justified in regarding the various de 
grees of mental activity from the simplest reflex 
manifestation to the highest mental act as a con 
nected series of increasingly manifold and pur 
posive actions proceeding from the same physico- 
chemical and physiological foundation. 

57. Feeling, Thinking, Acting. For good reasons 
it is generally assumed that the organisms have not 
always been what they are now, but have " de- 



Feeling, Thinking, Acting 175 

. veloped " from previous simpler forms. It is un 
decided whether originally there were one or sev 
eral forms from which the present forms sprang, 
nor is it known how life first made its appearance 
on earth. So long as the various assumptions with 
regard to this question have not led to decisive, actu 
ally demonstrable differences in the results, a dis- 

cussion of it is fruitless, and therefore unscientific. 
The usual word evolution is non-purposive in so 
far as it signifies the appearance of something al 
ready existing. Another conception is better ac 
cording to which the influence of changed conditions 
of existence has yielded the most important factor 
of change. 

The change that the organisms undergo is always 

in a definite direction. More and more complex 
and manifold forms are evolved, and the evolution 
of these forms is characterized by an ever greater 
specialization of the functions of life, so that every 
specially developed organ comes to perform but one 
function. It is true that by this means the organ 
ism is better fitted to perform those functions, but 
at the same time it grows more susceptible to injury, 

. since its existence depends upon the proper simul 
taneous activity of many different organs. Such 
an evolution, therefore, can occur only when the 
general conditions of life have grown steadier, so 
that the danger of disturbance becomes less. We 
are accustomed to regard changes in this direction 
as higher developments, and the progressive sim- 



176 The Biologic Sciences 

plifications of the organization (as for example in 
parasites) as backward steps. 

Since our opinion as to what constitutes a higher 
and a lower organism is doubtless arbitrary, let us 
ask whether it is not possible to find an objective 
standard by which to measure the relative perfec 
tion of the different organisms. The question must 
be answered in the affirmative when we take into 
consideration the following. Since the quantity of 
available free energy upon the earth is limited, the 
organism which transforms the energy at its dis 
posal more completely and with the least loss into 
the forms of energy necessary for the function of 
life, must be regarded as the more perfect organ 
ism. In fact, we observe that with increasing com 
plexity of the organisms there is for the most part 
also an increasing improvement in that direction, 
and we can therefore speak of some beings as more 
perfect than others. This view-point is especially 
significant in the evaluation of human progress, ap 
pearing, as it does, as the general standard of all civ 
ilization. 

The perfection of the organism shows itself in 

^relation to the outer world in the development of 
the sense organs. While a single-celled animal re 
acts almost exclusively to chemical, sometimes also 
to optical, stimuli, and receives these with the en- 

ttire surface of its body, special parts of the body 
develop more and more toward perfection. These 
are the parts that respond with special ease to the ap- 



Feeling, Thinking, Acting 177 

propriate stimuli, that is, react to them with an in- 

1 creasingly smaller expenditure of energy. Then the 
points at which the stimuli are received are separated 
from those in which the reaction occurs, and the two 
are connected by conducting paths, the nerves, in 
which an energy process takes place. Our present 
knowledge of this process still leaves much to be de 
sired. It is a process which moves with fairly great 
but by no means extraordinary rapidity (about ten to 
thirty meters per second) along the conducting 
paths. At the one end of this path it is caused by 
actions of various kinds, chiefly that of the specific 

*energy, for which the sense organ is developed. At 
the other end it discharges specific effects. There 
is no doubt that here we have in each instance a case 
of energy transformation connected with a discharge, 

t that is, with the action of other energies which lie 
at the ends ready for change. Hence there is no 
equivalence between the different kinds of energy, 

the discharging and the discharged, mostly not even 
a proportional relation, although both increase and 
decrease simultaneously. 

What the form of the energy is that is prop 
agated in the nerves is unknown. It can be either a 
special form which arises only under the conditions 
here present (just as, for example, a galvanic stream 
develops only under definite chemical and spacial 
conditions), or a special combination of known 
energies, as in sound and probably in light. Some 
day, it is likely, we shall have a more accurate knowl- 



178 The Biologic Sciences 

edge of the nerve process which will solve the ques 
tion. 

When such a process is caused by some energy 
impulse from without, it may produce various re 
sults. In the simplest case it discharges the corre- 

\ spending reaction, just as the leaves of the sensitive 
plant close when they are touched. Or it may give 
J rise to a series of processes following one another 
like the instinctive actions. Or, finally, it may 
bring about a series of inner processes which lead 
to an extreme differentiation of slight differences 
of this stimulus and to a corresponding graded re 
action with the anticipation of success. We call this 
conscious thinking, willing, and acting. 

Through the age-long effect of the blunder com 
mitted by Plato in making a fundamental distinc 
tion between mental life and physical life, we 
experience the utmost difficulty in habituating our 
selves to the thought of the regular connection be- 

: tween the simplest physiological and the highest in 
tellectual acts. Moreover, this contrast has been 
accentuated by the mechanical hypothesis. If we 
abandon the mechanical hypothesis and adhere to 
the summarization of experience free from all 
hypotheses, as represented in the science of energy, 
this contrast disappears. For even if we concede 
the impossibility of conceiving thought as mechan 
ical, there is no difficulty in conceiving of it as 

i energetic, especially since we know that mental work 
is connected with expenditure of energy and ex- 



Society 179 

haustion just as physical work is. However, the 
elucidation of this subject lies almost entirely in 
the future since the idea just developed has but 
only begun to influence scientific work in this 
field. But judging from the results that have 
already been obtained we may hope for a speedy\ 
development. 

58. Society. The external circumstance that as 
an organism multiplies the new being must come to 
life in the proximity of the older one, is in itself 
cause for the formation of closed groups confined to 
certain localities by animal organisms of the same 
species. But they become scattered if the advantage 
of their living together is not such as to outweigh 
the disadvantage of having a narrow field of com 
petition for the means of sustenance. Thus we see 
different plants and animals behaving differently in 
this respect. While some species live in as great 
isolation as possible, others form communities, even 
if there is no mechanical tie to hold them together 
by a common integument. 

Since the second case is true of man in a highly 
marked degree, his social characteristics and needs 
form a large and important part of his life. And 
jsince, further, the socialization of man makes con 
tinuous headway with increasing civilization we 
need but think of the development of the former 
little groups and tribes into states and the present 
very active internationalization of the most im 
portant affairs of mankind, especially of the sci- 



180 The Biologic Sciences 

ences the social problems also occupy an ever 
larger place in the organization of human life. 

What distinguishes man most essentially from the 
other animals, even the most advanced, is his 
capacity for perfection, which in the lower animal 
can be paralleled at best by its capacity for self- 
preservation. While the organization of the ani 
mals within the short period of which we have any 
historical knowledge has to all appearances re 
mained essentially unchanged, the world of man 
kind has changed in quite a remarkable way. This 
change consists in an increasing subjection of the 

* external world to human purposes, and rests upon 
the increasing socialization of his capacities. 

Memory and heredity (the latter being but an 
extension of memory to the offspring, which is to 
be conceived of as a part of the older organism) 
secures in the first place only the preservation of 
the stock and the renewed development of the new 
individual in the average type. If a specially fa 
vored individual succeeds in accomplishing greater 
achievements, he may in favorable circumstances 

transmit this capacity for higher attainments to his 
offspring. But such individuals gain an advantage 
in the struggle for existence only if the other sides 
of their activity do not suffer curtailment as a re 
sult. With the limited amount of energy at the in 
dividual s disposal every extraordinary accomplish- 

ment involves a corresponding one-sidedness, and 
as soon as a certain measure is slightly overstepped, 



Society 181 

it will cause a reduction of the other functions which 
will render the individual less fit in the struggle for 
existence. But this is true only so long as an in 
dividual must live by himself. As soon as he forms 
part of a social organization which benefits by his 
particular activity, the organization compensates for 
the personal disadvantages by its collective activity, 
and a social community not only finds room for 
such special developments, but it even encourages 
and promotes them. 

We have already seen that such manifestations 
occur within the organism itself. Higher functions, 
depending upon the higher susceptibility of the sense 
organs, can only be attained at the expense of the 
general functions by the organ in question. We 
observe this fact in all socially organized beings, like 
bees and ants, which display a high degree of spe 
cialization in the functions of the individual sub 
ordinate groups; the specialization often being car 
ried so far that the individual groups can no longer 
subsist by themselves alone. It is only the organi-1 
zation as a whole that is capable of permanent ex- \ 
istence. 

While the evolution of such superior functions 
involves a corresponding differentiation, and there 
fore a division and separation of the superior func 
tions within the social structure, the necessity for 
communication and for mutual support results in 
an approximation of the individuals and the groups. 
In every society, therefore, the centrifugal and the 



i8a The Biologic Sciences 

centripetal forces work simultaneously in co-opera 
tion and in opposition to one another. While the 
extreme specialization on the one hand seems to 
make for the best individual functioning, on the 
other hand it renders the entire collective structure 
much more dependent, and therefore much more 
subject to injury, as is shown by the example of the 
queen bee, whose departure threatens the existence 
of the entire hive. Thus a medium degree of dif 
ferentiation will as a general rule produce the most 
permanent social structure. 

59. Language and Intercourse. The essential 
value of the social organization resides in the fact 
that the work of the individual, in so far as it is 
s adapted to it, accrues to the benefit of the collective 
whole. For this it is absolutely essential that the 
members of the collectivity should be able to have 
intercourse with one another in order that every 
part of the general activity may be communicated 
to the others. This intercourse is obtained through 
language in the most general sense. 

We have already learned that the essence of lan- 
, guage consists in the co-ordination of concept to 
sign. The social application of language demands 
that the signs co-ordinated to the concepts in use 
should be the same for all the members of the so 
cial organization. Only in this way can the mem 
bers make themselves mutually understood. But 
intelligible means of communication and division of 
labor impart to the social knowledge that is set 



Language and Intercourse 183 

down in writing a kind of independent exist 
ence. Many centuries ago the possibility ceased for 
one person to store in his memory the entire stock 
of human knowledge. Nowadays we have men 
who are versed only in single parts of separate sci 
ences, and the aggregate knowledge appears at first 
to be a unity existing only in thought. But because 
this knowledge is set down in signs which endure far 
beyond the life of the individual and at the appropri 
ate moment can unfold its entire power even after a 
long period of inactivity, it has acquired an existence 
of a social character independent of the individual. 
For although it survives the individual, it cannot 
survive the death of human society. 

As the socialization of all mankind advances to 
ever greater unities, the linguistic limitations sprung 
(from former stages of evolution prove to be a 
hindrance. The mother tongue, of course, forms 
the first and most important entry for the individual 
to the common store of knowledge. But in view of 
the linguistic limitation of which I have just spoken 
the efforts in our day are carried on with renewed 
zeal to create a universal auxiliary language (p. 100) 
by means of which intercourse should be made pos 
sible beyond the language boundaries. There have 
already been gratifying results.* 

* At the present time " Ido " is the best. It is a highly 
practicable artificial language, and its advocates have succeeded 
in organizing it to insure its normal development. An older 
and still rather widespread form called " Esperanto " has 



184 The Biologic Sciences 

60. Civilization. Everything which serves the so 
cial progress of mankind is appropriately called civ 
ilization or culture, and the objective characteristic 
of progress consists in improved methods for seizing 
and utilizing the raw energies of nature for human 
purposes. Thus it was a cultural act when a primi 
tive man discovered that he could extend the radius 
of his muscle energy by taking a pole in his hand, 
and it was another cultural act when a primitive man 
discovered that by throwing a stone he could send 
his muscle energy a distance of many meters to the 
desired point. The effect of the knife, the spear, 
the arrow, and of all the other primitive implements 
can be called in each case a purposive transformation 
of energy. And at the other end of the scale of 
civilization the most abstract scientific discovery, by 
reason of its generalization and simplification, sig 
nifies a correspondng economy of energy for all the 
coming generations that may have anything to do 
with the matter. Thus, in fact, the concept of 
progress as here defined embraces the entire sweep 
of human endeavor for perfection, or the entire 
field of culture, and at the same time it shows the 
great scientific value of the concept of energy. 

If we consider further that, according to the sec 
ond fundamental principle, the free energy accessi 
ble to us can only decrease, but not increase, while 
the number of men whose existence depends directly 

failed to organize itself so as to insure its development and it 
must inevitably die out. 



Civilization 185 

on the consumption of a due amount of free energy 
is constantly on the increase, then we at once see 
the objective necessity of the development of civ 
ilization in that sense. His foresight puts man in 
a position to act culturally. But if we examine our 
present social order from this point of view, we 
realize with horror how barbarous it still is. Not 
only do murder and war destroy cultural values 
without substituting others in their place, not only 
do the countless conflicts which take place between 
the different nations and political organizations act 
anticulturally, but so do also the conflicts between 
the various social classes of one nation, for they de 
stroy quantities of free energy which are thus with 
drawn from the total of real cultural values. At 
present mankind is in a state of development in 
which progress depends much less upon the leader 
ship of a few distinguished individuals than upon 
the collective labor of all workers. Proof of this 
is that it is coming to be more and more the fact 
that the great scientific discoveries are made simul 
taneously by a number of independent investigators 
- an indication that society creates in several places i 
the individual conditions requisite for such discov 
eries. Thus we are living at a time when men are 
gradually approximating one another very closely 
in their natures, and when the social organization 
therefore demands and strives for as thorough an 
equalization as possible in the conditions of ex 
istence of all men. 



INDEX 



Above and below, distinction 
between, 121 

Abstract, concrete and, 16 ff. 

Abstraction, 20 

Action, conscious, 174; in 
stinctive, 174 

Adaptation, 172 ff. 

Aeromechanics, 147 

Algebra, 80 

Alikeness, definition of, 51 ff. 

Allotropic changes, 161 

Analysis, infinitesimal, in 

Analytic geometry, 122 ff. 

Analytic judgments, 66 

Anthropology, 57 

Ants, specialization of, 181 

Applied sciences, 57 ff. 

A priori judgments, 44 

Aristotle, 38, 66 

Aristotle s logic, 22 

Arithmetic, 79 ff. 

Assertions, never absolutely 
correct, 53 

Association, 63 ff., 81 

Astronomic objective, 6 

Astronomy as an applied sci 
ence, 58 

Atomic hypothesis in chemis 
try, 142 

Atoms, 141 

Bees, specialization of, 181 
Biological sciences, 55; life 

most general concept in, 

56 
Botany, 56 



Caesar, Julius, 76 
Capillary phenomena, 146 



187 



Capillary theory, 147 

Carbon, its circulation 
through plants and ani 
mals, 171 ; life based on the 
energy of, 168 

Carbonic acid, 171 

Carnot, Sadi, 151 

Causal relation, purification 
of, 34 ff- 

Causation, the law of, 31 ff. 

Chemical combinations, 71 ff. ; 
quantitative relations in, 74 

Chemical energy, 159 ff. ; 
capable of powerful con 
centration, 161 ; different 
forms of, 159 

Chemical formulas represent 
concepts not sounds, 95 

Chemistry, 20, 47, 55; signifi 
cance of, TOO ff. 

Chinese script based on di 
rect co-ordination, 93 

Civilization, 184 ff. 

Classification, not definite, 2; 
purpose of, 2-4 

Classification of the sciences, 
53 ff ; 

Collective activity, 181 

Combination, sequence in, 

73 ?. 

Combinations, theory of, 71 
Combinatory schematization, 

73; in chemistry, 71 ff. ; in 

physics, 72 
Communication, 181 
Community among plants and 

animals, 179 
Comparison, 82 
Comte, Auguste, 54 



1 88 



Index 



Concept, the most general, 
61 ff. 

Concepts, arbitrary, 23; com 
plex, 23; complex empiri 
cal, 23; definition of, 16; 
empirical, 18 ; formation 
of, 19 ; general, 26 ; in 
ceaseless flux, 88; science 
of, 15 ff., 122; simple, 20; 
simple and complex, 19 ff. 

Conclusion, the, 24 ff. ; ana 
lytic, 66; scientific, 27, 30, 
66 ff. 

Concrete and abstract, 16 ff. 

Conjugacy, most general con 
cept in formal sciences, 56 

Conscious action, 174 

Conscious thinking, willing, 
and acting, 178 

Conservation of energy, the 
law of the, 135 ff. 

Conservation of matter, 138 

Conservation of the sum of 
work and kinetic energy, 
the law of the, 134 

Conservation of work, the 
law of the, 130 

Conservation, quantitative, 

131 

Continuity, 101 ff. ; the law 
of, 113 ff. 

Co-ordinated signs, change 
in, 88 ff. 

Co-ordination, 80 ff. ; a means 
of obtaining facts without 
dealing directly with the 
corresponding realities, 87 ; 
between concept and word 
not unambiguous, 89; be 
tween concept and written 
sign, direct and indirect, 
92 ff. ; identity the limit 
case in, 82 ; integral num 
bers the best basis of, 85; 
in use among primitive 
men and higher animals, 
87; its importance, 85; 
methodology of the sci 



ences based upon, 85; of 

numbers with signs, 90 ff. ; 

possibility of unambiguous, 

88 

Copernican theory, 117 ff. 
Copernicus, 117, 141 
Corpuscular theory of light, 

5, 157 
Counting, 85 ff. ; defined, 85 ; 

purpose of, 86 
Culture, see Civilization 

Darwin, his fundamental the 
ory, 173 

Deduction, 40 ff. ; the process 
of, 41 ff. 

Deductive sciences, 38 

Determinateness, absolute, 
only in ideal world, 50 

Determinateness of things, 
the, 47 ff. 

Determinism, 48, 51 

Differential Calculus, see Dif 
ferentials 

Differentials, 112 

Double numbers or double 
points in a group, 82 

Dynamics, 128 ff. ; definition 
of, 139 

Elasticity, 145 

Elastic undulatory theory of 
light, see Wave theory of 
light 

Electricity, principal source 
of, 156 

Electricity and magnetism, 
154 ff. 

Electromagnetic theory of 
light, 157 ff- 

Electro-technics, 156 

Empirical sciences, 38 

Energy, a substance, 136; at 
rest, 154; free, 154; im 
portance of concept of, 
128; in nerves, 177; the 
most general concept in 
the physical sciences, 56; 



Index 



189 



of form, 145; of volume, 

145 

Energy intensity, 153 

Energetic mechanics, 138 ff. 

Erg, definition of, 150 

Esperanto, 183, note 

Euclid, 44, 140 

European-American scripts 
based on indirect co-ordi 
nation, 93 

Experience, incompleteness 
of, 27; more limited than 
the conceivable, 118 

Experiences, distinguished by 
being familiar, 31; limited 
knowledge of, 31 

Experiential sciences, see 
Empirical sciences 

Extrapolation, 46, 50, 104. 

Familiarity due to recalling 
former similar experiences, 
ii 

Fechner, 102 

Feeling, thinking, acting, 
174 ff- 

Force, 129 ff., 153 

Formal sciences, 54 ; are em 
pirical sciences, 55; order 
most general concept in, 56 

Foucault s pendulum experi 
ment, 121 

Freedom of the will, 50 ff. 

Frequency of process facili 
tates repetition, 11 ff. 

Function, 109 ff. ; continuous 
and discontinuous, no; 
most general concept in 
formal sciences, 56 

Functional relation, the appli 
cation of the, 112 ff. 

Functions, the theory of, in 

Fundamental principle, the 
second, 150 ff. 

Gases, 145 

Generalization, suitable place 
for, in text-books, 9 ff. 



Geometry, 47, 54, 119, 127; 
ancient and modern meth 
ods of, no ff. 
Goethe, 99 

Good usage in language, 100 
Grammatical correctness, im 
portance attached to, 99 
Grammatical rules, 97 
Gravitation potential, the, 112 
Group, the, 65 ff. ; double 
members or double points 
in, 82; linear arrangement 
of members of, 75 ff. 
Groups, artificial and natural, 
69 ff. ; closed, among ani 
mals, 179; infinite, equality 
of, 84; related, 69 ff. ; un 
equivocal order of, 83 

Heat, mechanical equivalent 
of, 149; theory of, 147 ff. 

Heat energy, 148 ff. 

Heat engine, 151 ; ideal, 
151 ff. 

Heat quantity, 148 ff. 

Heliotrope, 90 

Herbart, 102 

Heredity, 180 

Higher analysis, in 

Homonyn, 89 

Hydromechanics, 147 

Ideal cases, 44 ff. 

Ideal machines, 132 

Identity, the limit case in co 
ordination, 82 

Ido, 183, note 

Imperfection, indestructible 
quality of science, 4 

Incompleteness, no hindrance 
to efficiency of science, 5 

Indestructibility of matter, 
see Conservation of mat 
ter 

Indo-Arabic notation, 91 

Induction, 38; the complete 
and the incomplete, 39 

Inductive sciences, 38 



190 



Index 



Inference, by induction, 38; 

from analogy, 140 
Infinitesimal analysis, HI 
Inorganic world, lack of 

memory and foresight in, 

33 

Insoluble problems, 142 
Instinctive action, 174 
Intercourse, language and, 

182 ff. 
Isolation among plants and 

animals, 179 
Isomeric, 74 
Isomeric changes, 161 

Judgments, analytic, 66 

Kant, 44, 66, 105 

Kepler, 141 

Kinetic energy, 132; and 
work, their sum constant, 
133 ff. ; transformed into 
work and vice versa, 134 

Knowledge, aim of, 19; in 
dividual, compared to tele 
phone, 7 ff. ; limited, 31; 
possibility of error in, in 
eradicable, 40; social char 
acter of, 183 

Language, beginnings of, 88; 
defective in co-ordination, 
96; distinction between sci 
ence and knowledge of, 98 ; 
good usage in, 100; 
and intercourse, 182 ff. ; 
needless inflections in, 99 ff. ; 
of words more imperfect 
than written language, 92; 
purpose of its cultivation, 
99; the science of, 97 ff. ; 
unambiguity the ideal of, 
89; a universal auxiliary, 
TOO; written, 89 ff. 

Leibnitz, 88; his doctrine of 
pre-established harmony, 
143; inventor of differen 
tials, 112 



Life, 163 ff. ; the most gen 
eral concept in the biolog 
ical sciences, 56 

Light, 5, 156 ff. 

Liquids, 145 

Locke, John, 21 ff., 88; his 
elaboration of the notion 
of simple and complex 
" ideas," 21 ; his secondary 
qualities, 127 

Logic, 54, 67 ff. ; content of, 
19; definition of, 15 ff. 

Luther, 99 

Magnetism, electricity and, 

154 ff- 
Man, compared to pair of 

sieves, 34; his capacity for 

perfection, 180 
Manifold, the science of the, 

54 

Mass, 132 ff., 136 ff. ; a sub 
stance, 138 
Mathematical laws, accuracy 

of, 105 

Mathematics, 54; an empiri 
cal science, 55 ; influence 
on, of concept of continu 
ity, in; its progress after 
introduction of Indo-Ara- 
bic numerals and algebraic 
signs, 101 

Matter, definition of, 138 
Mayer, Julius Robert, 149; 
his discovery of the law of 
conservation, 151 
Measurement, 107 
Mechanical energies, 144 
Mechanics, 55, 128 ff. ; com 
plementary branches of, 
144 ff. ; definition of, 138; 
early development of, 139; 
energetic, 138 ff. ; the first 
branch of physics treated 
mathematically, 139 ; pure 
or classical, 144 
Mechanistic hypothesis, the, 
as an interpretation of all 



Index 



191 



natural phenomena, 142 ; 
especially injurious in study 
of mental phenomena, 142 

Mechanistic theories, 140 ff. 

Mechanistic theory of the 
universe, 132 

Mechanization of astronomy, 
141 

Memory, 16, 32, 180; defi 
nition of, 172; general 
characteristic of, 61 ; lack 
of, in inorganic world, 

53 

Metabolism, 165 
Methodology of the sciences 
based upon co-ordination, 

?S 

Microscope, 6 

Motion, the science of, 54, 

122 ; uninfluenced, 122 
Musical notation, 93 

Names, arbitrariness of, 62; 
signs and, 86 ff. 

Natural laws, 28 ff. ; defini 
tion of, 28; their extent 
dependent upon stage of 
knowledge in each sci 
ence, 7 ; their usual origin, 
42 ff. ; prediction from, 
only approximate, 48 

Natural philosophy, defini 
tion of, i ; importance of, 
in study of science, 10; 
place of, in text-books, 
9ff. 

Negation, 68 ff. 

Nerves, 177 

Nervous discharge, 177 

Newton, Sir Isaac, 141 

Number groups, unlimited, 
78 

Numbers, 78 ff. ; theory of, 
80 

Numerals, co-ordination of, 
with signs, 86 

Numerical names different in 
different languages, 86 



Numerical signs internation 
al, 86 
Nutrition, 165 

Objective, astronomic, 6; 
photographic, 6 

Objective character of the 
world, 34 

Optical telegraph, 90 

Optics, geometric, 5 

Optic signs, 90 

Order, most general concept 
in formal sciences, 56 

Organisms, standard for mea 
suring relative perfection 
of, 176; stationary forms, 
163 

Orthography, efforts to im 
prove, 99; English, defec 
tive in co-ordination, 96; 
exaggerated importance of 
correctness in, 99 ; mistakes 
in, 97; reform of, 97 

Parabolic curve, 48 
Paradoxes of the infinite, 

84 
Pasigraphy, 92 ff. ; Chinese 

system of, 94 
Permanent in change, the, 

131 

Perpetual motion, 130 
Perpetual motion machine, 

153 

Philology, 97 ff. 
Philosophy, limited progress 

in, 101 

Phonetic writing, 33 ff. 
Phoronomy, 54, 119, 122, 127 
Photographic objective, 6. 
Photo chemical processes, 

foundation of terrestial 

life, 169 

Physical sciences, 55 
Physics, 47, 55 ; the science 

of the different kinds of 

energy, 72; each branch of, 



192 



Index 



tfeats of a special kind of 
energy, 139 

Physiology, 55 ff. 

Plato, his distinction between 
mental and physical life, 
178 

Polarity of electricity and 
magnetism, 155 

Political organizations, con 
flicts between, 185 

Prediction, 12 

Pre-established harmony, 

143 

Pressure, 146, 154 

Progress, depends on collec 
tive labor, 185 ; economy 
of energy, 184; evaluation 
of, 176 

Pure science, 57 

Pseudo-problems in science, 
142 

Psychology, 47, 55 ff. 

Psycho-physical parallelism, 

143 
Ptolemy s system, 117 

Quantity, the science of, see 
Mathematics, 54 

Radiant energy, 157; its im 
portance to man, 158 

Rational sciences, see Deduc 
tive sciences 

Rays, straight lines of, 5 

Reaction, teleological, 173 

Reality, 16 ff. 

Reflection, 5 

Reflex action, 173 

Refraction, 5 

Repetition, basis of conscious 
life, 174 

Reproduction, 165 ff. 

Roman notation, 91 

Science, aim of, 13 ff. ; com 
parison of, to a network, 
42; comparison of, to a 
tree or forest, 6; definition 



of, 13; eternal truth of, 
6 ff. ; " for its own sake," 
13 ff. ; the facts of, unal 
terable, 8 ff. ; the function 
of, 23, 37; importance of 
theoretical, 15; its pro 
cedure, 45; the study of 
happiness, 28 

Sciences, the table of the> 
54 ff. 

Scientific discoveries, inde 
pendent simultaneous, 185 

Scientific instinct, 43 

Scientific materialism, 138 

Scientific written language 
based on direct co-ordina 
tion, 93 

Self-preservation, 180 

Sense organs, 176 ff. 

Shakespeare, 99 

Signs and names, 86 ff. 

Social characteristics, im 
portance of, 179 ff. 

Social classes, conflicts be 
tween, 185 

Socialization of human ca 
pacities, 180 

Social order still barbarous, 
185 

Social organization, 180 ; how 
best obtained, 182 ; its ten 
dency to equalize condi 
tions, 185; secures perma 
nence among specialized 
individuals, 181 

Social problems, 179 ff. 

Society, 179 ff. ; centrifugal 
and centripetal forces in, 
181 ff. ; division of func 
tions in, 181 

Sociology, 47, 55, 57 

Solar radiation, 169 

Soul, the, 171 ff. 

Sound signs, advantage and 
disadvantage of, 89 ff. 

Sound writing, 33 ff., 92 ff. 

Space, four-dimensional, 77, 
note; homogeneity of, in 



Index 



193 



horizontal direction, 121 ; 
the science of, 54; sym 
metrical and tri-dimension- 
al, 118; time and, 118 ff. ; 
tri-dimensional, 76 

Specialization, one-sidedness 
of, 180 ff. 

Spelling reform, 97 

Stable forms, 163 

Statics, 128 ff. ; definition of, 
138 ff. 

Stationary bodies, capable of 
regeneration, 164 

Stationary forms, 163 

Substance, 132 

Surface-energy, 146 

Syllogism, the, classic method 
of argumentation, 65 ff. 

Synonym, 89 

Table of the sciences, 54 ff. 

" Teleological " properties of 
organisms, 173 

Teleological reaction, 173 

Telegraph, optical, 90 

Telescope, 5 

Temperature, 148 

Theoretical science, impor 
tance of, 15 

Theory of functions, in 

Theory of numbers, 80 

Thermo-chemistry, 37 

Thermo-dynamics, 153 



Thing, definition of, 62 ff. 

Thought conceived of as en 
ergetic, 178 

Threshold, 102 

Time, a form of inner life, 
76; measurement of, 122; 
one-seried, or one-dimen 
sional, 118; and space, 
118 ff. 

Unambiguity, in language, 
89; of co-ordination of 
numbers to signs, 90 

Universal auxiliary 1 a n - 
guage, 100, 183 

Velocity, 133 
Volume energy, 145 

War, 185 

Wave surface, 6 

Wave theory of light, 5, 157 

Weight, 132, 137 ff. ; a sub 
stance, 138 

Work, mechanical, 129; prod 
uct of the force and the 
distance, 130; a substance 
in a limited sense, 136 

Written language, 89 ff. 

Written signs, 90 

Zoology, 56 



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Journal of a Russian Prisoner s Wife in Japan. Illustrated 
from photographs. $1.50 net, by mail $1.62. 

" Holds a tremendous human interest. . . . Author writes with wit 
and a delightfully feminine abandon." Outlook. 

"This surprisingly outspoken volume . , . could have been written 
only by an extraordinarily able woman who knew the inside of Russian 
politics and also had actual experience in Japanese war hospitals." Chicago 
Record-Herald. 

W. F. JOHNSON S FOUR CENTURIES OF THE PANAMA 
CANAL 

With 16 illustrations and 6 colored maps. $3.00 net ; by mail, 
$3.27. 

" The most thorough and comprehensive book on the Panama Canal." 
Nation. 

JOHN L. GIVENS MAKING A NEWSPAPER 

The author was recently with the New York Evening Sun. 
$1.50 net; by mail $1.62. 

Some seventy-five leading newspapers praise this book as the 
best detailed account of the business, editorial, reportorial and 
manufacturing organization of a metropolitan journal. It should 
be invaluable to those entering upon newspaper work and a 
revelation to the general reader. 

THE OPEN ROAD THE FRIENDLY TOWN 

Compiled by E. V. Lucas. Full gilt, illustrated cover linings, 
each (cloth) $1.50 ; (leather) $2.50. 

Pretty anthologies of prose and verse from British and 
American authors, respectively for wayfarers and the urbane. 

* If the reader will send his name and address the publishers will send, 
from time to time, information regarding their new books. 

HENRY HOLT AND COMPANY 

PUBLISHERS (X- 07) NEW TOBK 



RECENT VOLUMES IN 

THE AMERICAN NATURE SERIES 

(Prospectus on request) 

SHELL-FISH INDUSTRIES B * 

Large I2mo. Illustrated by half-tones and original draw 
ings. Just published. $1.75 net. 

Covers classification, propagation and distribution. For the person 
who eats oysters, clams or scallops, there is information on their 
structure, life-histories and habits. A chapter is devoted to shell-fish 
as collectors and carriers of disease organisms. The oyster culturist 
will find the life history of bivalves, a comparison of various culture 
methods, and a description of oyster fields in various parts of the 
world. Several facts concerning the habits of bivalves, here presented 
for the first time, will be of interest to naturalists. 

FISH STORIES: Alleged and Experienced, with a Little 
History, Natural and Unnatural 

By CHARLES F. HOLDER, Author of " The Log of a Sea 
Angler," etc., and DAVID STARR JORDAN, Author of " A Guide 
to the Study of Fishes," etc. With colored plates and many 
illustrations from photographs. $1.75 net. 

" A delightful miscellany, telling about fish of the strangest kind, 
with scientific description melting into accounts of personal adventure. 
Nearly everything that is entertaining in the fish world is touched upon 
and science and fishing are made very readable." New York Sun. 

INSECT STORIES By VERNON L. KELLOGG. 

Illustrated, $1.50 net. 

Strange, true stories, primarily for children, but certainly 
for those grown-ups who like to read discriminatingly to their 
children. 

" The author is among a few scientific writers of distinction who 
can interest the popular mind. No intelligent youth can fail to read 
it with delight and profit." The Nation. 

THE LIFE OF A FOSSIL HUNTER CHARLS H By STERNBERG> 

With introduction by PROF. H. F. OSBORN, 48 Illustrations, 

$1.60 net. 

The most interesting autobiography of the oldest and best 
known explorer in this field. 

" One of the most interesting books to be found anywhere." William 
Allen White. 

THE FRESH WATER AQUARIUM By OTTO EGGELING and 
AND ITS INHABITANTS FREDERICK EHRENBERG. 

A Guide for the Amateur Aquarist. With 100 illustrations, 
large I2mo, $2.00 net. 

" The best guide to the aquarium." The Independent. 

HENRY HOLT AND COMPANY 

PUBLISHERS NEW YORK 



THE THEORY OF THE THEATRE 

And Other Principles of Dramatic Criticism 

By CLAYTON HAMILTON. Author of " Materials and Methods 
of Fiction." $1.50 net; by mail, $1.60. 

CONTENTS : 

THE THEORY OF THE THHATRE. What is a Play? The Psychology 
of Theatre Audiences. The Actor and the Dramatist. Stage Con 
ventions in Modern Times. Economy of Attention in Theatrical Per 
formances. Emphasis in the Drama. The Four Leading Types of 
Drama: Tragedy and Melodrama; Comedy and Farce. The Modern 
Social Drama, 

OTHER PRINCIPLES OF DRAMATIC CRITICISM. The Public and the 
Dramatist. Dramatic Art and the Theatre Business. The Happy End 
ings in the Theatre. The Boundaries of Approbation. Imitation and 
Suggestion in the Drama. Holding the Mirror up to Nature. Blank 
Verse on the Contemporary Stage. Dramatic Literature and Theatric 
journalism. The Intention of Performance. The Quality of New 
Endeavor. The Effect of Plays upon the Public. Pleasant and Un 
pleasant Plays. Themes in the Theatre. The Function of Imagination. 

DRAMATISTS OF TO-DAY 

ROSTAND, HAUPTMANN, SUDERMANN, 
PINERO, SHAW, PHILLIPS, MAETERLINCK 

By PROF. EDWARD EVERETT HALE, JR., of Union College. With 
gilt top, $1.50 net. (By mail, $1.60.) 

An informal discussion of their principal plays and of the perform 
ances of some of them. The volume opens with a paper " On Stand 
ards of Criticism," and concludes with " Our Idea of Tragedy," and 
an appendix of all the plays of each author, with dates of their first 
performance or publication. 

New York Evening Post: " It is not often nowadays that a theat 
rical book can be met with so free from gush and mere eulogy, or so 
weighted by common sense ... an excellent chronological appendix 
and full index . . . uncommonly useful for reference." 

Dial: "Noteworthy example of literary criticism in one of the 
most interesting of literary fields. . . . Well worth reading a second 
time." 

THE GERMAN DRAMA OF THE 
NINETEENTH CENTURY 

By GEORG WITKOWSKI. Translated by PROF. L. E. HORNING. 
I2mo. $1.00. 



Kleist, Grillparzer, Hebbel, Ludwig, Wildenbruch, Sudermann, Haupt- 
mann, and minor dramatists receive attention. 

New York Times Review: "The translation of this brief, clear, and 
logical account was an extremely happy idea. Nothing at the same time 
so comprehensive and terse has appeared on the subject, and it is a 
subject of increasing interest to the English-speaking public." 



HENRY HOLT AND COMPANY 

PUBLISHERS NEW YORK 





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